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21 Bootstrapping Regression Models B ootstrapping is a nonparametric approach to statistical inference that substitutes computation for more traditional distributional assumptions and asymptotic results.1 Bootstrapping offers a number of advantages: • The bootstrap is quite general, although there are some cases in which it fails. • Because it does not require distributional assumptions (such as normally distributed errors), the bootstrap can provide more accurate inferences when the data are not well behaved or when the sample size is small. • It is possible to apply the bootstrap to statistics with sampling distributions that are difficult to derive, even asymptotically. • It is relatively simple to apply the bootstrap to complex data-collection plans (such as stratified and clustered samples). 21.1 Bootstrapping Basics My principal aim is to explain how to bootstrap regression models (broadly construed to include generalized linear models, etc.), but the topic is best introduced in a simpler context: Suppose that we draw an independent random sample from a large population.2 For concreteness and simplicity, imagine that we sample four working, married couples, determining in each case the husband's and wife's income, as recorded in Table 21.1. I will focus on the difference in incomes between husbands and wives, denoted as Yi for the ith couple. We want to estimate the mean difference in income between husbands and wives in the pop- ulation. Please bear with me as I review some basic statistical theory: A point estimate of this population mean difference μ is the sample mean, Yi 6−3+5+3 Y = = = 2.75 n 4 Elementary statistical theory tells us that the standard deviation of the sampling distribution of √ sample means is SD(Y ) = σ/ n, where σ is the population standard deviation of Y . If we knew σ , and if Y were normally distributed, then a 95% confidence interval for μ would be σ μ = Y ± 1.96 √ n 1 The term bootstrapping, coined by Efron (1979), refers to using the sample to learn about the sampling distribution of a statistic without reference to external assumptions—as in "pulling oneself up by one's bootstraps." 2 In an independent random sample, each element of the population can be selected more than once. In a simple random sample, in contrast, once an element is selected into the sample, it is removed from the population, so that sampling is done "without replacement." When the population is very large in comparison to the sample (say, at least 10 times as large), the distinction between independent and simple random sampling becomes inconsequential. 587 588 Chapter 21. Bootstrapping Regression Models Table 21.1 Contrived "Sample" of Four Married Couples, Showing Husbands' and Wives' Incomes in Thousands of Dollars Observation Husband's Income Wife's Income Difference Yi 1 24 18 6 2 14 17 −3 3 40 35 5 4 44 41 3 where z.025 = 1.96 is the standard normal value with a probability of .025 to the right. If Y is not normally distributed in the population, then this result applies asymptotically. Of course, the asymptotics are cold comfort when n = 4. In a real application, we do not know σ . The standard estimator of σ is (Yi − Y )2 S= n−1 from which the standard error of the mean (i.e., the estimated standard deviation of Y ) is SE(Y ) = √ S/ n. If the population is normally distributed, then we can take account of the added uncertainty associated with estimating the standard deviation of the mean by substituting the heavier-tailed t-distribution for the normal distribution, producing the 95% confidence interval S μ = Y ± tn−1, .025 √ n Here, tn−1, .025 is the critical value of t with n − 1 degrees of freedom and a right-tail probability of .025. √ In the present case, S = 4.031, SE(Y ) = 4.031/ 4 = 2.015, and t3, .025 = 3.182. The 95% confidence interval for the population mean is thus μ = 2.75 ± 3.182 × 2.015 = 2.75 ± 6.41 or, equivalently, −3.66 < μ < 9.16 As one would expect, this confidence interval—which is based on only four observations—is very wide and includes 0. It is, unfortunately, hard to be sure that the population is reasonably close to normally distributed when we have such a small sample, and so the t-interval may not be valid.3 Bootstrapping begins by using the distribution of data values in the sample (here, Y1 = 6, Y2 = −3, Y3 = 5, Y4 = 3) to estimate the distribution of Y in the population.4 That is, we define the random variable Y ∗ with distribution5 3 To say that a confidence interval is "valid" means that it has the stated coverage. That is, a 95% confidence interval is valid if it is constructed according to a procedure that encloses the population mean in 95% of samples. 4 An alternative would be to resample from a distribution given by a nonparametric density estimate (see, e.g., Silverman & Young, 1987). Typically, however, little if anything is gained by using a more complex estimate of the population distribution. Moreover, the simpler method explained here generalizes more readily to more complex situations in which the population is multivariate or not simply characterized by a distribution. 5 The asterisks on p ∗ (·), E ∗ , and V ∗ remind us that this probability distribution, expectation, and variance are conditional on the specific sample in hand. Were we to select another sample, the values of Y1 , Y2 , Y3 , and Y4 , would change and— along with them—the probability distribution of Y ∗ , its expectation, and variance. 21.1. Bootstrapping Basics 589 y∗ p∗ (y∗ ) 6 .25 −3 .25 5 .25 3 .25 Note that E ∗ (Y ∗ ) = y ∗ p(y ∗ ) = 2.75 = Y all y∗ and V ∗ (Y ∗ ) = [y ∗ − E ∗ (Y ∗ )]2 p(y ∗ ) 3 2 n−1 2 = 12.187 = S = S 4 n Thus, the expectation of Y ∗ is just the sample mean of Y , and the variance of Y ∗ is [except for the factor (n − 1)/n, which is trivial in larger samples] the sample variance of Y . We next mimic sampling from the original population by treating the sample as if it were the population, enumerating all possible samples of size n = 4 from the probability distribution of Y ∗ . In the present case, each bootstrap sample selects four values with replacement from among the four values of the original sample. There are, therefore, 44 = 256 different bootstrap samples,6 each selected with probability 1/256. A few of the 256 samples are shown in Table 21.2. Because the four observations in each bootstrap sample are chosen with replacement, particular bootstrap samples usually have repeated observations from the original sample. Indeed, of the illustrative bootstrap samples shown in Table 21.2, only sample 100 does not have repeated observations. ∗ ∗ ∗ ∗ ∗ Let us denote the bth bootstrap sample7 as yb = [Yb1 , Yb2 , Yb3 , Yb4 ] , or more generally, yb∗ = [Y ∗ , Y ∗ , . . . , Y ∗ ] , where b = 1, 2, . . . , nn . For each such bootstrap sample, we b1 b2 bn calculate the mean, n ∗ ∗ i=1 Ybi Yb = n The sampling distribution of the 256 bootstrap means is shown in Figure 21.1. The mean of the 256 bootstrap sample means is just the original sample mean, Y = 2.75. The standard deviation of the bootstrap means is nn ∗ ∗ ∗ b=1 (Y b − Y )2 SD (Y ) = nn = 1.745 We divide here by nn rather than by nn − 1 because the distribution of the nn = 256 bootstrap sample means (Figure 21.1) is known, not estimated. The standard deviation of the bootstrap 6 Many of the 256 samples have the same elements but in different order—for example, [6, 3, 5, 3] and [3, 5, 6, 3]. We could enumerate the unique samples without respect to order and find the probability of each, but it is simpler to work with the 256 orderings because each ordering has equal probability. 7 If vector notation is unfamiliar, then think of y∗ simply as a list of the bootstrap observations Y ∗ for sample b. b bi 590 Chapter 21. Bootstrapping Regression Models Table 21.2 A Few of the 256 Bootstrap Samples for the Data Set [6, −3, 5, 3], and the Corresponding ∗ Bootstrap Means, Yb Bootstrap Sample ∗ b Y∗ b1 Y∗ b2 Y∗ b3 Y∗ b4 Yb 1 6 6 6 6 6.00 2 6 6 6 −3 3.75 3 6 6 6 5 5.75 . . . . . . . . . 100 −3 5 6 3 2.75 101 −3 5 −3 6 1.25 . . . . . . . . . 255 3 3 3 5 3.50 256 3 3 3 3 3.00 means is nearly equal to the usual standard error of the sample mean; the slight slippage is due to √ the factor n/(n − 1), which is typically negligible (though not when n = 4):8 n ∗ SE(Y ) = SD∗ (Y ) n−1 4 2.015 = × 1.745 3 This precise relationship between the usual formula for the standard error and the bootstrap standard deviation is peculiar to linear statistics (i.e., linear functions of the data) like the mean. For the mean, then, the bootstrap standard deviation is just a more complicated way to calculate what we already know, but • bootstrapping might still provide more accurate confidence intervals, as I will explain presently; and • bootstrapping can be applied to nonlinear statistics for which we do not have standard-error formulas or for which only asymptotic standard errors are available. Bootstrapping exploits the following central analogy: The population is to the sample as the sample is to the bootstrap samples. Consequently, • ∗ the bootstrap observations Ybi are analogous to the original observations Yi ; ∗ • the bootstrap mean Y b is analogous to the mean of the original sample Y ; • the mean of the original sample Y is analogous to the (unknown) population mean μ; and • the distribution of the bootstrap sample means is analogous to the (unknown) sampling distribution of means for samples of size n drawn from the original population. 8 See Exercise 21.1. 21.1. Bootstrapping Basics 591 Y = 2.75 25 20 Frequency 15 10 5 0 −2 0 2 4 6 Bootstrap Mean Figure 21.1 Graph of the 256 bootstrap means from the sample [6, −3, 5, 3]. The broken vertical line gives the mean of the original sample, Y = 2.75, which is also the mean of the 256 bootstrap means. Bootstrapping uses the sample data to estimate relevant characteristics of the population. The sampling distribution of a statistic is then constructed empirically by resampling from the sample. The resampling procedure is designed to parallel the process by which sample observations were drawn from the population. For example, if the data represent an independent random sample of size n (or a simple random sample of size n from a much larger population), then each bootstrap sample selects n observations with replacement from the original sample. The key bootstrap analogy is the following: The population is to the sample as the sample is to the bootstrap samples. The bootstrapping calculations that we have undertaken thus far depend on very small sample size, because the number of bootstrap samples (nn ) quickly becomes unmanageable: Even for samples as small as n = 10, it is impractical to enumerate all the 1010 = 10 billion bootstrap samples. Consider the "data" shown in Table 21.3, an extension of the previous example. The Y mean and standard deviation of the differences in income√ are Y = 4.6 and S = 5.948. Thus, the standard error of the sample mean is SE(Y ) = 5.948/ 10 = 1.881. Although we cannot (as a practical matter) enumerate all the 1010 bootstrap samples, it is easy to draw at random a large number of bootstrap samples. To estimate the standard deviation of a statistic (here, the mean)—that is, to get a bootstrap standard error—100 or 200 bootstrap samples should be more than sufficient. To find a confidence interval, we will need a larger number of bootstrap samples, say 1,000 or 2,000.9 A practical bootstrapping procedure, therefore, is as follows: 1. Let r denote the number of bootstrap replications—that is, the number of bootstrap samples to be selected. 9 Results presented by Efron and Tibshirani (1993, chap. 19) suggest that basing bootstrap confidence intervals on 1,000 bootstrap samples generally provides accurate results, and using 2,000 bootstrap replications should be very safe. 592 Chapter 21. Bootstrapping Regression Models Table 21.3 Contrived "Sample" of 10 Married Couples, Showing Husbands' and Wives' Incomes in Thousands of Dollars Difference Observation Husband's Income Wife's Income Yi 1 24 18 6 2 14 17 −3 3 40 35 5 4 44 41 3 5 24 18 6 6 19 9 10 7 21 10 11 8 22 30 −8 9 30 23 7 10 24 15 9 ∗ ∗ 2. For each bootstrap sample b = 1, . . . , r, randomly draw n observations Yb1 , Yb2, . . . , ∗ with replacement from among the n sample values, and calculate the bootstrap sample Ybn mean, n ∗ ∗ i=1 Ybi Yb = n 3. From the r bootstrap samples, estimate the standard deviation of the bootstrap means:10 r ∗ ∗ 2 ∗ b=1 Yb − Y SE∗ (Y ) = r −1 where r ∗ b=1 Y b ∗ Y ≡ r ∗ is the mean of the bootstrap means. We can, if we wish, "correct" SE∗ (Y ) for degrees of √ freedom, multiplying by n/(n − 1). To illustrate this procedure, I drew r = 2, 000 bootstrap samples, each of size n = 10, from ∗ the "data" given in Table 21.3, calculating the mean, Y b , for each sample. A few of the 2,000 bootstrap replications are shown in Table 21.4, and the distribution of bootstrap means is graphed in Figure 21.2. We know from statistical theory that were we to enumerate all the 1010 bootstrap samples (or, alternatively, to sample infinitely from the population of bootstrap samples), the average bootstrap ∗ mean would be E ∗ (Y ) = Y = 4.6, and the standard deviation of the bootstrap means would be ∗ 10 It is important to distinguish between the "ideal" bootstrap estimate of the standard deviation of the mean, SD∗ (Y ), ∗ which is based on all nn bootstrap samples, and the estimate of this quantity, SE∗ (Y ), which is based on r randomly selected bootstrap samples. By making r large enough, we seek to ensure that SE ∗ (Y ∗ ) is close to SD∗ (Y ∗ ). Even ∗ SD∗ (Y ) = SE(Y ) is an imperfect estimate of the true standard deviation of the sample mean SD(Y ), however, because it is based on a particular sample of size n drawn from the original population. 21.1. Bootstrapping Basics 593 Table 21.4 A Few of the r = 2, 000 Bootstrap Samples Drawn From the Data Set [6, −3, 5, 3, 6, 10, 11, −8, 7, 9] and the Corresponding ∗ Bootstrap Means, Yb ∗ b Y∗ b1 Y∗ b2 Y∗ b3 Y∗ b4 Y∗ b5 Y∗ b6 Y∗ b7 Y∗ b8 Y∗ b9 Y∗ b10 Yb 1 6 10 6 5 −8 9 9 6 11 3 5.7 2 9 9 7 7 3 3 −3 −3 −8 6 3.0 3 9 −3 6 5 10 6 10 10 10 6 6.9 . . . . . . . . . 1999 6 9 6 3 11 6 6 7 3 9 6.6 2000 7 6 7 3 10 6 9 3 10 6 6.7 Y = 4.6 200 150 Frequency 100 50 0 −2 0 2 4 6 8 10 Bootstrap Mean Figure 21.2 Histogram of r = 2,000 bootstrap means, produced by resampling from the "sample" [6, −3, 5, 3, 6, 10, 11, −8, 7, 9]. The heavier broken vertical line gives the sample mean, Y = 4.6; the ligher broken vertical lines give the boundaries of the 95% percentile confidence interval for the population mean μ based on the 2,000 bootstrap samples. The procedure for constructing this confidence interval is described in the next section. ∗ √ √ SE∗ (Y ) =SE(Y ) (n − 1)/n = 1.881 9/10 = 1.784. For the 2,000 bootstrap samples that I ∗ ∗ selected, Y = 4.693 and SE(Y ) = 1.750—both quite close to the theoretical values. The bootstrapping procedure described in this section can be generalized to derive the empirical sampling distribution for an estimator θ of the parameter θ: 1. Specify the data-collection scheme S that gives rise to the observed sample when applied to the population:11 S(Population) ⇒ Sample 11 The "population" can be real—the population of working married couples—or hypothetical—the population of conceivable replications of an experiment. What is important in the present context is that the sampling procedure can be described concretely. 594 Chapter 21. Bootstrapping Regression Models The estimator θ is some function s(·) of the observed sample. In the preceding example, the data-collection procedure is independent random sampling from a large population. 2. Using the observed sample data as a "stand-in" for the population, replicate the data- collection procedure, producing r bootstrap samples: ⎧ ⎪ ⇒ Bootstrap sample1 ⎪ ⎪ ⎨ ⇒ Bootstrap sample2 S(Sample) . ⎪ ⎪ . . ⎪ ⎩ ⇒ Bootstrap sampler ∗ 3. For each bootstrap sample, calculate the estimate θb = s(Bootstrap sampleb ). ∗ 4. Use the distribution of the θb s to estimate properties of the sampling distribution of θ . For example, the bootstrap standard error of θ is SE∗ (θ ∗ ) (i.e., the standard deviation of the r ∗ bootstrap replications θb ):12 r ∗ ∗ b=1 (θb − θ )2 SE∗ (θ ∗ ) ≡ r −1 where r ∗ ∗ b=1 θb θ ≡ r 21.2 Bootstrap Confidence Intervals 21.2.1 Normal-Theory Intervals As I have mentioned, normal-theory confidence intervals for means are based on the t-distribution when the population variance of Y is unknown. Most statistics, including sam- ple means, are asymptotically normally distributed; in large samples we can therefore use the bootstrap standard error, along with the normal distribution, to produce a 100(1 − a)% confi- dence interval for θ based on the estimator θ: θ = θ ± za/2 SE∗ (θ ∗ ) (21.1) In Equation 21.1, za/2 is the standard normal value with probability a/2 to the right. This approach will work well if the bootstrap sampling distribution of the estimator is approximately normal, and so it is advisable to examine a normal quantile-comparison plot of the bootstrap distribution. There is no advantage to calculating normal-theory bootstrap confidence intervals for linear statistics like the mean, because in this case the ideal bootstrap standard deviation of the statistic and the standard error based directly on the sample coincide. Using bootstrap resampling in this setting just makes for extra work and introduces an additional small random component into standard errors. √ 12 We may want to apply the correction factor n/(n − 1). 21.2. Bootstrap Confidence Intervals 595 ∗ Having produced r bootstrap replicates θb of an estimator θ, the bootstrap standard error is r ∗ ∗ the standard deviation of the bootstrap replicates: SE∗ (θ ∗ ) = b=1 (θb − θ )2 /(r − 1), ∗ ∗ where θ is the mean of the θb . In large samples, where we can rely on the normality of θ, a 95% confidence interval for θ is given by θ ± 1.96 SE∗ (θ ∗ ). 21.2.2 Percentile Intervals Another very simple approach is to use the quantiles of the bootstrap sampling distribution of the estimator to establish the end points of a confidence interval nonparametrically. Let θ(b) ∗ represent the ordered bootstrap estimates, and suppose that we want to construct a (100 − a)% confidence interval. If the number of bootstrap replications r is large (as it should be to construct ∗ ∗ a percentile interval), then the a/2 and 1 − a/2 quantiles of θb are approximately θ(lower) and ∗ θ(upper) , where lower = ra/2 and upper = r(1 − a/2). If lower and upper are not integers, then ∗ we can interpolate between adjacent ordered values θ(b) or round off to the nearest integer. A nonparametric confidence interval for θ can be constructed from the quantiles of the ∗ bootstrap sampling distribution of θ ∗ . The 95% percentile interval is θ(lower) < θ < ∗ ∗ θ(upper) , where the θ(b) are the r ordered bootstrap replicates; lower = .025 × r and upper = .975 × r. A 95% confidence interval for the r = 2, 000 resampled means in Figure 21.2, for example, is constructed as follows: lower = 2000(.05/2) = 50 upper = 2000(1 − .05/2) = 1950 ∗ Y (50) = 0.7 ∗ Y (1950) = 7.8 0.7 < μ < 7.8 The end points of this interval are marked in Figure 21.2. Because of the skew of the bootstrap distribution, the percentile interval is not quite symmetric around Y = 4.6. By way of comparison, the standard t-interval for the mean of the original sample of 10 observations is μ = Y ± t9, .025 SE(Y ) = 4.6 ± 2.262 × 1.881 = 4.6 ± 4.255 0.345 < μ < 8.855 In this case, the standard interval is a bit wider than the percentile interval, especially at the top. 596 Chapter 21. Bootstrapping Regression Models 21.2.3 Improved Bootstrap Intervals I will briefly describe an adjustment to percentile intervals that improves their accuracy.13 As before, we want to produce a 100(1 − a)% confidence interval for θ having computed the sample ∗ estimate θ and bootstrap replicates θb ; b = 1, . . . , r. We require za/2 , the unit-normal value with probability a/2 to the right, and two "correction factors," Z and A, defined in the following manner: • Calculate ⎡ r ⎤ ∗ # (θb < θ) −1 ⎢ b=1 ⎥ Z≡ ⎣ ⎦ r ∗ where −1 (·) is the inverse of the standard normal distribution function, and #(θb < θ)/r is the proportion of bootstrap replicates below the estimate θ . If the bootstrap sampling distribution is symmetric and if θ is unbiased, then this proportion will be close to .5, and the "correction factor" Z will be close to 0. • Let θ(−i) represent the value of θ produced when the ith observation is deleted from the sample;14 there are n of these quantities. Let θ represent the average of the θ(−i) ; that is, θ ≡ n θ(−i) /n. Then calculate i=1 n i=1 (θ(−i) − θ) 3 A≡ n (21.2) i=1 (θ(−i) − θ) ] 6[ 2 3/2 With the correction factors Z and A in hand, compute Z − za/2 A1 ≡ Z+ 1 − A(Z − za/2 ) Z + za/2 A2 ≡ Z+ 1 − A(Z + za/2 ) where (·) is the cumulative standard normal distribution function. Note that when the correction factors Z and A are both 0, A1 = (−za/2 ) = a/2, and A2 = (za/2 ) = 1 − a/2. The values A1 and A2 are used to locate the end points of the corrected percentile confidence interval. In particular, the corrected interval is ∗ ∗ θ(lower*) < θ < θ(upper*) where lower* = rA1 and upper* = rA2 (rounding or interpolating as required). The lower and upper bounds of percentile confidence intervals can be corrected to improve the accuracy of these intervals. 13 The interval described here is called a "bias-corrected, accelerated" (or BCa ) percentile interval. Details can be found in Efron and Tibshirani (1993, chap. 14); also see Stine (1990) for a discussion of different procedures for constructing bootstrap confidence intervals. 14 The θ (−i) are called the jackknife values of the statistic θ . The jackknife values can also be used as an alternative to the bootstrap to find a nonparametric confidence interval for θ . See Exercise 21.2. 21.3. Bootstrapping Regression Models 597 Applying this procedure to the "data" in Table 21.3, we have z.05/2 = 1.96 for a 95% confidence interval. There are 926 bootstrapped means below Y = 4.6, and so Z = −1 (926/2000) = −0.09288. The Y (−i) are 4.444, 5.444, . . . , 4.111; the mean of these values is Y = Y = 4.6,15 and (from Equation 21.2) A = −0.05630. Using these correction factors, −0.09288 − 1.96 A1 = −0.09288 + 1 − [−.05630(−0.09288 − 1.96)] = (−2.414) = 0.007889 −0.09288 + 1.96 A2 = −0.09288 + 1 − [−.05630(−0.09288 + 1.96)] = (1.597) = 0.9449 Multiplying by r, we have 2000 × .0.007889 ≈ 16 and 2000 × .0.9449 ≈ 1890, from which ∗ ∗ Y (16) < μ < Y (1890) (21.3) −0.4 < μ < 7.3 Unlike the other confidence intervals that we have calculated for the "sample" of 10 differences in income between husbands and wives, the interval given in Equation 21.3 includes 0. 21.3 Bootstrapping Regression Models The procedures of the previous section can be easily extended to regression models. The most straightforward approach is to collect the response-variable value and regressors for each observation zi ≡ [Yi , Xi1 , . . . , Xik ] Then the observations z1 , z2 , . . . , zn can be resampled, and the regression estimator computed for ∗ ∗ ∗ each of the resulting bootstrap samples, zb1 , zb2 , . . . , zbn , producing r sets of bootstrap regression coefficients, bb∗ = [A∗ , B ∗ , . . . , B ∗ ] . The methods of the previous section can be applied to b b1 bk compute standard errors or confidence intervals for the regression estimates. Directly resampling the observations zi implicitly treats the regressors X1 , . . . , Xk as random rather than fixed. We may want to treat the Xs as fixed (if, e.g., the data derive from an experimental design). In the case of linear regression, for example, 1. Estimate the regression coefficients A, B1 , . . . , Bk for the original sample, and calculate the fitted value and residual for each observation: Yi = A + B1 xi1 + · · · + Bk xik Ei = Yi − Yi ∗ ∗ ∗ ∗ 2. Select bootstrap samples of the residuals, eb = [Eb1 , Eb2 , . . . , Ebn ] , and from these, ∗ = [Y ∗ , Y ∗ , . . . , Y ∗ ] , where Y ∗ = Y + E ∗ . calculate bootstrapped Y values, yb b1 b2 bn bi i bi 15 The average of the jackknifed estimates is not, in general, the same as the estimate calculated for the full sample, but this is the case for the jackknifed sample means. See Exercise 21.2. 598 Chapter 21. Bootstrapping Regression Models 3. Regress the bootstrapped Y values on the fixed X values to obtain bootstrap regression coefficients. *If, for example, estimates are calculated by least-squares regression, then b∗ = b ∗ (X X)−1 X yb for b = 1, . . . , r. ∗ ∗ 4. The resampled b∗ = [A∗ , Bb1 , . . . , Bbk ] can be used in the usual manner to construct b b bootstrap standard errors and confidence intervals for the regression coefficients. Bootstrapping with fixed X draws an analogy between the fitted value Y in the sample and the conditional expectation of Y in the population, and between the residual E in the sample and the error ε in the population. Although no assumption is made about the shape of the error ∗ distribution, the bootstrapping procedure, by constructing the Ybi according to the linear model, implicitly assumes that the functional form of the model is correct. Furthermore, by resampling residuals and randomly reattaching them to fitted values, the procedure implicitly assumes that the errors are identically distributed. If, for example, the true errors have nonconstant variance, then this property will not be reflected in the resampled residuals. Likewise, the unique impact of a high-leverage outlier will be lost to the resampling.16 Regression models and similar statistical models can be bootstrapped by (1) treating the regressors as random and selecting bootstrap samples directly from the observations zi = [Yi , Xi1 , . . . , Xik ] or (2) treating the regressors as fixed and resampling from the residuals Ei of the fitted regression model. In the latter instance, bootstrap observations ∗ ∗ are constructed as Ybi = Yi + Ebi , where the Yi are the fitted values from the origi- ∗ nal regression, and the Ebi are the resampled residuals for the bth bootstrap sample. In ∗ each bootstrap sample, the Ybi are then regressed on the original Xs. A disadvantage of fixed-X resampling is that the procedure implicitly assumes that the functional form of the regression model fit to the data is correct and that the errors are identically distributed. To illustrate bootstrapping regression coefficients, I will use Duncan's regression of occupa- tional prestige on the income and educational levels of 45 U. S. occupations.17 The Huber M estimator applied to Duncan's regression produces the following fit, with asymptotic standard errors shown in parentheses beneath each coefficient:18 Prestige = −7.289 + 0.7104 Income + 0.4819 Education (3.588) (0.1005) (0.0825) Using random resampling, I drew r = 2, 000 bootstrap samples, calculating the Huber esti- mator for each bootstrap sample. The results of this computationally intensive procedure are summarized in Table 21.5. The distributions of the bootstrapped regression coefficients for income and education are graphed in Figure 21.3(a) and (b), along with the percentile confidence intervals for these coefficients. Figure 21.3(c) shows a scatterplot of the bootstrapped coefficients 16 For these reasons, random-X resampling may be preferable even if the X values are best conceived as fixed. See Exercise 21.3. 17 These data were discussed in Chapter 19 on robust regression and at several other points in this text. 18 M estimation is a method of robust regression described in Section 19.1. 21.4. Bootstrap Hypothesis Tests* 599 Table 21.5 Statistics for r = 2, 000 Bootstrapped Huber Regressions Applied to Duncan's Occupational Prestige Data Coefficient Constant Income Education Average bootstrap estimate −7.001 0.6903 0.4918 Bootstrap standard error 3.165 0.1798 0.1417 Asymptotic standard error 3.588 0.1005 0.0825 Normal-theory interval (−13.423,−1.018) (0.3603,1.0650) (0.2013,0.7569) Percentile interval (−13.150,−0.577) (0.3205,1.0331) (0.2030,0.7852) Adjusted percentile interval (−12.935,−0.361) (0.2421,0.9575) (0.2511,0.8356) NOTES: Three bootstrap confidence intervals are shown for each coefficient. Asymptotic standard errors are also shown for comparison. for income and education, which gives a sense of the covariation of the two estimates; it is clear that the income and education coefficients are strongly negatively correlated.19 The bootstrap standard errors of the income and education coefficients are much larger than the asymptotic standard errors, underscoring the inadequacy of the latter in small samples. The simple normal-theory confidence intervals based on the bootstrap standard errors (and formed as the estimated coefficients ±1.96 standard errors) are reasonably similar to the percentile intervals for the income and education coefficients; the percentile intervals differ slightly from the adjusted ∗ ∗ ∗ percentile intervals. Comparing the average bootstrap coefficients A , B 1 , and B 2 with the corre- sponding estimates A, B1 , and B2 suggests that there is little, if any, bias in the Huber estimates.20 21.4 Bootstrap Hypothesis Tests* In addition to providing standard errors and confidence intervals, the bootstrap can also be used to test statistical hypotheses. The application of the bootstrap to hypothesis testing is more or less obvious for individual coefficients because a bootstrap confidence interval can be used to test the hypothesis that the corresponding parameter is equal to any specific value (typically 0 for a regression coefficient). More generally, let T ≡ t (z) represent a test statistic, written as a function of the sample z. The contents of z vary by context. In regression analysis, for example, z is the n × k + 1 matrix [y, X] containing the response variable and the regressors. For concreteness, suppose that T is the Wald-like test statistic for the omnibus null hypothesis H0 : β1 = · · · = βk = 0 in a robust regression, calculated using the estimated asymptotic covariance matrix for the regression coefficients. That is, let V11 contain the rows and columns (k×k) of the estimated asymptotic covariance matrix V(b) that pertain to the k slope coefficients b1 = [B1 , . . . , Bk ] . We can write the null hypothesis as H0 : β1 = 0. Then the test statistic is −1 T = b1 V11 b1 19 The negative correlation of the coefficients reflects the positive correlation between income and education (see Section 9.4.4). The hint of bimodality in the distribution of the income coefficient suggests the possible presence of influential observations. See the discussion of Duncan's regression in Section 4.6. 20 For the use of the bootstrap to estimate bias, see Exercise 21.4. 600 Chapter 21. Bootstrapping Regression Models (a) (b) 350 350 250 250 Frequency Frequency 150 150 50 50 0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Income Coefficient Education Coefficient (c) 0.8 Education Coefficient 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Income Coefficient Figure 21.3 Panels (a) and (b) show histograms and kernel density estimates for the r = 2,000 bootstrap replicates of the income and education coefficients in Duncan's occupational prestige regression. The regression model was fit by M estimation using the Huber weight function. Panel (c) shows a scatterplot of the income and education coefficients for the 2,000 bootstrap samples. We could compare the obtained value of this statistic to the quantiles of χk , but we are loath to 2 do so because we do not trust the asymptotics. We can, instead, construct the sampling distribution of the test statistic nonparametrically, using the bootstrap. ∗ ∗ Let Tb∗ ≡ t (zb ) represent the test statistic calculated for the bth bootstrap sample, zb . We have to be careful to draw a proper analogy here: Because the original-sample estimates play the role of the regression parameters in the bootstrap "population" (i.e., the original sample), the bootstrap analog of the null hypothesis—to be used with each bootstrap sample—is H0 : β1 = B1 , . . . , βk = Bk . The bootstrapped test statistic is, therefore, ∗−1 Tb∗ = (b∗ − b1 ) Vb,11 (b∗ − b1 ) b1 b1 21.5. Bootstrapping Complex Sampling Designs 601 Having obtained r bootstrap replications of the test statistic, the bootstrap estimate of the p-value for H0 is simply21 r #b=1 (Tb∗ ≥ T ) p∗ = r Note that for this chi-square-like test, the p-value is entirely from the upper tail of the distribution of the bootstrapped test statistics. Bootstrap hypothesis tests proceed by constructing an empirical sampling distribution for the test statistic. If T represents the test statistic computed for the original sample, and Tb∗ is the test statistic for the bth of r bootstrap samples, then (for a chi-square-like test statistic) the p-value for the test is #(Tb∗ ≥ T )/r. 21.5 Bootstrapping Complex Sampling Designs One of the great virtues of the bootstrap is that it can be applied in a natural manner to more complex sampling designs. If, for example, the population is divided into S strata, with ns observations drawn from stratum s, then bootstrap samples can be constructed by resampling ns observations with replacement from the sth stratum. Likewise, if observations are drawn into the sample in clusters rather than individually, then the bootstrap should resample clusters rather than individuals. We can still calculate estimates and test statistics in the usual manner using the bootstrap to assess sampling variation in place of the standard formulas, which are appropriate for independent random samples but not for complex survey samples. When different observations are selected for the sample with unequal probabilities, it is com- mon to take account of this fact by differentially weighting the observations in inverse proportion to their probability of selection.22 Thus, for example, in calculating the (weighted) sample mean of a variable Y , we take n (w) i=1 wi Yi Y = n i=1 wi and to calculate the (weighted) correlation of X and Y , we take (w) wi (Xi − X)(Yi − Y ) rXY = [ wi (Xi − X)2 ][ wi (Yi − Y )2 ] Other statistical formulas can be adjusted analogously.23 The case weights are often scaled so that wi = n, but simply incorporating the weights in the usual formulas for standard errors does not produce correct results. Once more, the bootstrap (0) 21 There is a subtle point here: We use the sample estimate b1 in place of the hypothesized parameter β 1 to calculate the bootstrapped test statistic Tb∗ regardless of the hypothesis that we are testing—because in the central bootstrap analogy b1 stands in for β 1 (and the bootstrapped sampling distribution of the test statistic is computed under the assumption that the hypothesis is true). See Exercise 21.5 for an application of this test to Duncan's regression. 22 These "case weights" are to be distinguished from the variance weights used in weighted-least-squares regression (see Section 12.2.2). 23 See Exercise 21.6. 602 Chapter 21. Bootstrapping Regression Models provides a straightforward solution: Draw bootstrap samples in which the probability of inclusion is proportional to the probability of inclusion in the original sample, and calculate bootstrap replicates of the statistics of interest using the case weights. The essential "trick" of using the bootstrap in these (and other) instances is to resample from the data in the same way as the original sample was drawn from the population. Statistics are calculated for each bootstrap replication in the same manner as for the original sample. The bootstrap can be applied to complex sampling designs (involving, e.g., stratification, clustering, and case-weighting) by resampling from the sample data in the same manner as the original sample was selected from the population. Social scientists frequently analyze data from complex sampling designs as if they originate from independent random samples (even though there are often non-negligible dependencies among the observations) or employ ad hoc adjustments (e.g., by weighting). A tacit defense of common practice is that to take account of the dependencies in complex sampling designs is too difficult. The bootstrap provides a simple solution.24 21.6 Concluding Remarks If the bootstrap is so simple and of such broad application, why isn't it used more in the social sciences? Beyond the problem of lack of familiarity (which surely can be remedied), there are, I believe, three serious obstacles to increased use of the bootstrap: 1. Common practice—such as relying on asymptotic results in small samples or treating dependent data as if they were independent—usually understates sampling variation and makes results look stronger than they really are. Researchers are understandably reluctant to report honest standard errors when the usual calculations indicate greater precision. It is best, however, not to fool yourself, regardless of what you think about fooling others. 2. Although the conceptual basis of the bootstrap is intuitively simple and although the cal- culations are straightforward, to apply the bootstrap it is necessary to write or find suitable statistical software. There is some bootstrapping software available, but the nature of the bootstrap—which adapts resampling to the data-collection plan and statistics employed in an investigation—apparently precludes full generality and makes it difficult to use traditional statistical computer packages. After all, researchers are not tediously going to draw 2,000 samples from their data unless a computer program can fully automate the process. This impediment is much less acute in programmable statistical computing environments.25 3. Even with good software, the bootstrap is computationally intensive. This barrier to boot- strapping is more apparent than real, however. Computational speed is central to the 24 Alternatively, we can use sampling-variance estimates that are appropriate to complex survey samples—a subject that is beyond the scope of this book. See, for example, Skinner, Holt, and Smith (1989). 25 See, for example, the bootstrapping software for the S (R and S-PLUS) statistical-computing environment described by Efron and Tibshirani (1993, app.) and by Davison and Hinkley (1997, chap. 11). Bootstrapping facilities are also provided in the Stata programming environment. Exercises 603 exploratory stages of data analysis: When the outcome of one of many small steps imme- diately affects the next, rapid results are important. This is why a responsive computing environment is especially useful for regression diagnostics, for example. It is not nearly as important to calculate standard errors and p-values quickly. With powerful, yet relatively inexpensive, desktop computers, there is nothing to preclude the machine from cranking away unattended for a few hours (although that is rarely necessary—a few minutes is more typical). The time and effort involved in a bootstrap calculation are usually small compared with the totality of a research investigation—and are a small price to pay for accurate and realistic inference. Exercises Exercise 21.1. ∗ Show that the mean of the nn bootstrap means is the sample mean nn ∗ ∗ b=1 Y b E ∗ (Y ) = =Y nn and that the standard deviation (standard error) of the bootstrap means is nn ∗ ∗ ∗ b=1 (Y b − Y )2 S SE (Y ) = =√ nn n−1 n where S = i=1 (Yi − Y ) /(n − 1) is the sample standard deviation. (Hint: Exploit the fact 2 that the mean is a linear function of the observations.) Exercise 21.2. The jackknife: The "jackknife" (suggested for estimation of standard errors by Tukey, 1958) is an alternative to the bootstrap that requires less computation, but that often does not perform as well and is not quite as general. Efron and Tibshirani (1993, chap. 11) show that the jackknife is an approximation to the bootstrap. Here is a brief description of the jackknife for the estimator θ of a parameter θ : 1. Divide the sample into m independent groups. In most instances (unless the sample size is very large), we take m = n, in which case each observation constitutes a "group." If the data originate from a cluster sample, then the observations in a cluster should be kept together. 2. Recalculate the estimator omitting the j th group, j = 1, . . . , m, denoting the resulting value of the estimator as θ(−j ) . The pseudo-value associated with the j th group is defined as ∗ θj ≡ mθ − (m − 1)θ(−j ) . 3. The average of the pseudo-values, θ ∗ ≡ ( m=1 θj )/m is the jackknifed estimate of θ . A j ∗ jackknifed 100(1 − a)% confidence interval for θ is given by S∗ θ = θ ∗ ± ta/2,m−1 √ n where ta/2,m−1 is the critical value of t with probability a/2 to the right for m − 1 degrees of m ∗ freedom, and S ∗ ≡ j =1 (θj − θ ∗ )2 /(m − 1) is the standard deviation of the pseudo-values. (a) ∗ Show that when the jackknife procedure is applied to the mean with m = n, the pseudo- values are just the original observations, θi∗ = Yi ; the jackknifed estimate θ ∗ is, therefore, 604 Chapter 21. Bootstrapping Regression Models the sample mean Y ; and the jackknifed confidence interval is the same as the usual t confidence interval. (b) Demonstrate the results in part (a) numerically for the contrived "data" in Table 21.3. (These results are peculiar to linear statistics like the mean.) (c) Find jackknifed confidence intervals for the Huber M estimator of Duncan's regression of occupational prestige on income and education. Compare these intervals with the bootstrap and normal-theory intervals given in Table 21.5. Exercise 21.3. Random versus fixed resampling in regression: (a) Recall (from Chapter 2), Davis's data on measured and reported weight for 101 women engaged in regular exercise. Bootstrap the least-squares regression of reported weight on measured weight, drawing r = 1, 000 bootstrap samples using (1) random-X resampling and (2) fixed-X resampling. In each case, plot a histogram (and, if you wish, a density estimate) of the 1,000 bootstrap slopes, and calculate the bootstrap estimate of standard error for the slope. How does the influential outlier in this regression affect random resampling? How does it affect fixed resampling? (b) Randomly construct a data set of 100 observations according to the regression model Yi = 5 + 2xi + εi , where xi = 1, 2, . . . , 100, and the errors are independent (but seriously heteroscedastic), with εi ∼ N (0, xi2 ). As in (a), bootstrap the least-squares regression of Y on x, using (1) random resampling and (2) fixed resampling. In each case, plot the bootstrap distribution of the slope coefficient, and calculate the bootstrap estimate of standard error for this coefficient. Compare the results for random and fixed resampling. For a few of the bootstrap samples, plot the least-squares residuals against the fitted values. How do these plots differ for fixed versus random resampling? (c) Why might random resampling be preferred in these contexts, even if (as is not the case for Davis's data) the X values are best conceived as fixed? Exercise 21.4. Bootstrap estimates of bias: The bootstrap can be used to estimate the bias of ∗ an estimator θ of a parameter θ, simply by comparing the mean of the bootstrap distribution θ (which stands in for the expectation of the estimator) with the sample estimate θ (which stands in ∗ for the parameter); that is, bias = θ − θ . (Further discussion and more sophisticated methods are described in Efron and Tibshirani, 1993, chap. 10.) Employ this approach to estimate the bias of the maximum-likelihood estimator of the variance, σ 2 = (Yi − Y )2 /n, for a sample of n = 10 observations drawn from the normal distribution N (0, 100). Use r = 500 bootstrap replications. How close is the bootstrap bias estimate to the theoretical value −σ 2 /n = −100/10 = −10? Exercise 21.5. ∗ Test the omnibus null hypothesis H0 : β1 = β2 = 0 for the Huber M estimator in Duncan's regression of occupational prestige on income and education. (a) Base the test on the estimated asymptotic covariance matrix of the coefficients. (b) Use the bootstrap approach described in Section 21.4. Exercise 21.6. Case weights: (a) ∗ Show how case weights can be used to "adjust" the usual formulas for the least-squares coefficients and their covariance matrix. How do these case-weighted formulas compare with those for weighted-least-squares regression (discussed in Section 12.2.2)? Summary 605 (b) Using data from a sample survey that employed disproportional sampling and for which case weights are supplied, estimate a least-squares regression (1) ignoring the case weights; (2) using the case weights to estimate both the regression coefficients and their standard errors (rescaling the case weights, if necessary, so that they sum to the sample size); and (3) using the case weights but estimating coefficient standard errors with the bootstrap. Compare the estimates and standard errors obtained in (1), (2), and (3). Exercise 21.7. ∗ Bootstrapping time-series regression: Bootstrapping can be adapted to time series regression but, as in the case of fixed-X resampling, the procedure makes strong use of the model fit to the data—in particular, the manner in which serial dependency in the data is modeled. Suppose that the errors in the linear model y = Xβ + ε follow a first-order autoregressive process (see Chapter 16), εi = ρεi−1 + υi ; the υi are independently and identically distributed with zero expectations and common variance συ . Suppose further that we use the method of 2 maximum likelihood to obtain estimates ρ and β. From the residuals e = y − Xβ, we can estimate υi as Vi = Ei − ρEi−1 for i = 2, . . . , n; by convention, we take V1 = E1 . Then, for each bootstrap replication, we sample n values with replacement from the Vi ; call them Vb1 , ∗ ∗ , . . . , V ∗ . Using these values, we construct residuals E ∗ = V ∗ and E ∗ = ρE ∗ Vb2 ∗ bn b1 b1 bi b, i−1 + Vbi for i = 2, . . . , n; and from these residuals and the original fitted values Yi = xi β, we construct ∗ ∗ ∗ bootstrapped Y values, Ybi = Yi + Ebi . The Ybi are used along with the original xi to obtain ∗ bootstrap replicates βb of the ML coefficient estimates. (Why are the xi treated as fixed?) Employ this procedure to compute standard errors of the coefficient estimates in the time-series regression for the Canadian women's crime-rate data (discussed in Chapter 16), using an AR(1) process for the errors. Compare the bootstrap standard errors with the usual asymptotic standard errors. Which standard errors do you prefer? Why? Then describe a bootstrap procedure for a time-series regression model with AR(2) errors, and apply this procedure to the Canadian women's crime-rate regression. Summary • Bootstrapping is a broadly applicable, nonparametric approach to statistical inference that substitutes intensive computation for more traditional distributional assumptions and asymp- totic results. The bootstrap can be used to derive accurate standard errors, confidence intervals, and hypothesis tests for most statistics. • Bootstrapping uses the sample data to estimate relevant characteristics of the popula- tion. The sampling distribution of a statistic is then constructed empirically by resampling from the sample. The resampling procedure is designed to parallel the process by which sample observations were drawn from the population. For example, if the data represent an independent random sample of size n (or a simple random sample of size n from a much larger population), then each bootstrap sample selects n observations with replacement from the original sample. The key bootstrap analogy is the following: The population is to the sample as the sample is to the bootstrap samples. ∗ • Having produced r bootstrap replicates θb of an estimator θ , the bootstrap standard error is the standard deviation of the bootstrap replicates: r ∗ ∗ ∗ ∗ − θ )2 b=1 (θb SE (θ ) = r −1 606 Chapter 21. Bootstrapping Regression Models ∗ ∗ where θ is the mean of the θb . In large samples, where we can rely on the normality of θ , a 95% confidence interval for θ is given by θ ± 1.96 SE∗ (θ ∗ ). • A nonparametric confidence interval for θ can be constructed from the quantiles of the ∗ ∗ bootstrap sampling distribution of θ ∗ . The 95% percentile interval is θ(lower) < θ < θ(upper) , where the θ(b)∗ are the r ordered bootstrap replicates; lower = .025×r and upper = .975×r. • The lower and upper bounds of percentile confidence intervals can be corrected to improve the accuracy of these intervals. • Regression models can be bootstrapped by (1) treating the regressors as random and selecting bootstrap samples directly from the observations zi = [Yi , Xi1 , . . . , Xik ] or (2) treating the regressors as fixed and resampling from the residuals Ei of the fitted regression model. In ∗ ∗ the latter instance, bootstrap observations are constructed as Ybi = Yi + Ebi , where the Yi are the fitted values from the original regression, and the Ebi ∗ are the resampled residuals for ∗ the bth bootstrap sample. In each bootstrap sample, the Ybi are then regressed on the original Xs. A disadvantage of fixed-X resampling is that the procedure implicitly assumes that the regression model fit to the data is correct and that the errors are identically distributed. • Bootstrap hypothesis tests proceed by constructing an empirical sampling distribution for the test statistic. If T represents the test statistic computed for the original sample and Tb∗ is the test statistic for the bth of r bootstrap samples, then (for a chi-square-like test statistic) the p-value for the test is #(Tb∗ ≥ T )/r. • The bootstrap can be applied to complex sampling designs (involving, e.g., stratification, clustering, and case weighting) by resampling from the sample data in the same manner as the original sample was selected from the population. Recommended Reading Bootstrapping is a rich topic; the presentation in this chapter has stressed computational procedures at the expense of a detailed account of statistical properties and limitations. • Although Efron and Tibshirani's (1993) book on the bootstrap contains some relatively advanced material, most of the exposition requires only modest statistical background and is eminently readable. • Davison and Hinkley (1997) is another statistically sophisticated, comprehensive treatment of bootstrapping. • A briefer source on bootstrapping addressed to social scientists is Stine (1990), which includes a fine discussion of the rationale of bootstrap confidence intervals. • Young's (1994) paper and the commentary that follows it focus on practical difficulties in applying the bootstrap.
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Free forum : This is for all the people who like Bleach and want to RPG or just disscuss. :D
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Click here to visit Northeast Georgia Marketplace, which includes classifieds from Dawson Community News, The Times of Gainesville and other regional papers.
Click here to place your ad for Dawson Community News with our sister paper, The Times, or call 770-535-1199 to speak with a representative..
A longtime employee accused of taking more than $100,000 from a local grading company turned herself into authorities last week.
Facing criminal charges for reportedly writing bad checks, the embattled director of a local sports academy was denied bond last week.
The director of a sports academy was extradited Tuesday to Dawson County, where he faces mounting criminal charges for reportedly writing thousands of dollars in bad checks.
A local accountant arrested two years ago on theft charges has been indicted again by a Dawson County grand jury.
Two of three local soldiers charged with tossing training grenades into a crowd at a Dawson County parking lot this summer have been released on bond.
A Big Canoe woman is out on bond after authorities said she tried to smother her 94-year-old mother-in-law with a pillow at her home last week.
A 19-year-old man from Hall County is in custody on a felony charge after he allegedly ran from authorities at the scene of a one-vehicle wreck in southern Dawson County.
As much as $100,000 in merchandise is reportedly missing from a local outlet store following a remodeling in mid-August.
A former assistant manager of a local retail store was sentenced Friday to more than seven years in federal prison on child pornography charges.
Domestic terrorism charges have been dropped against two local soldiers accused of tossing training grenades into a crowd last month.
A 53-year-old Dawson County man was arrested Monday on charges that he molested a young boy.
Two people face drug charges after an undercover sting Friday afternoon at a local convenience store.
An attorney for one of three Dahlonega soldiers accused of tossing grenade simulators into a Dawsonville crowd earlier this month said his client cannot pay the $200,000 bond a judge set Aug. 19.
Rhonda Randolph is a drug addict.
Charges likely will be filed in connection with a three-vehicle wreck on Hwy. 53 that sent three people to area hospitals Aug. 18.
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A woman involved in a fender bender with a deputy in 2013 has now filed suit against Dawson County.
A Dawsonville man has been charged with aggravated assault in connection with the brutal beating of a family member.
A Dahlonega man pleaded guilty last week to his part in a string of church burglaries in Lumpkin County.
Parents caught allowing their kids to host drinking parties with their friends could face hefty fines and punishments if a proposed ordinance is passed by county commissioners.
Authorities are investigating the thefts of guns and ammo taken from a deputy's service vehicle last week.
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I am selling my desk and chair from college. The desk is a solid oak, three drawer desk with a keyboard tray. The chair is from Ikea; red in color with a mesh back and fabric seat. It is fully adjustable and compact-perfect for the student. Desk and Chair sold together for $100
Desk: bought in 2011 for $125; for sale separately for $75
Chair: bought new in 2011 for $65; for sale separately ...
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Super Monkey Ball
440 reviews
By installing you agree to the Terms of Use and other terms
PublisherSega Networks Inc
More from Sega Networks Inc
Download size42 MB
Last updated2/12/2015
Works with
- Windows Phone 8.1
- Windows Phone 8
- Windows Phone 7.5
App requires
- Xbox
- photo, music, and video libraries
- data services
- movement and directional sensor
- HD720P (720x1280)
- WVGA (480x800)
- WXGA (768x1280)
Supported languages (11)
Deutsch English Español (España, alfabetización internacional) suomi français italiano 日本語 português (Brasil) русский 中文(简体) 中文(繁體)
Let's roll! Guide a cute little monkey encased in a transparent ball to victory by tilting and rolling him through a variety of slopes and turns. You must navigate a large variety of obstacle courses to reach goal lines within set time limits. Collect as many bananas as you can along the way and you'll be able to increase your remaining lives. Play through five worlds and 110 stages, designed to be fun for both first-time players and long-time Super Monkey Ball fans. Enjoy rich 3D graphics with an easy-to-master control system that utilizes the accelerometer inside your device. Insanely fun and addicting!
6/19/2015Hyvä peli!!!!!
6/16/2015Too bad
5/29/2015Curse that game!!!!! It isn't even worth $1!!!!!!!!
5/6/2015This game is a gimmick at best. Too sensitive controls render it nearly unplayable, making the game an achievement hunter's nightmare. These issues takes away all the fun from the experience. If it had MOGA support it'd redeem itself, I guess (which Sonic CD and 4 Episode 1 mysteriously lack, BTW). One star for graphics and another for attempting something new. I bought this just to support SEGA on Windows Phone, but avoid if you're not a fan of the company. Apparently Toshihiro Nagoshi only worked well under the supervision of Yu Suzuki, because the only good game he pulled off was Daytona USA, during the good old AM2 days. Sadly, everything else from him is just an unpolished mess, Yakuza included.
4/5/2015Fue genial
8/23/2014Has tilt controls like the 3DS super monkey ball. However, you can not calibrate controls, and flat on table = not move.
8/14/2014Paskaa ku maksaa. Mutta kokeiluversio hyvä
8/11/2014Its so mast
7/27/2014Delete super monkey game


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im surrounded by hypocrites
Question Posted Sunday April 29 2007, 7:30 pm
im surrounded my hypocrites. my cousin was talking shit about me to my best friend. well my "best friend" was "defending " me but in a way, also talking shit, saying that im always wrong and they are always right. they are calling me stupid when in reality, im smarter than both of them combined. Then I see that they are all of a sudden best friends. Im not a mean person and i have a good heart...so i can never tell when someone is being a hypocrite or not...plz help me...cause write now im crying and i have no one to talk to... :(
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answered Sunday April 29 2007, 8:20 pm:
one word foget them they aint worth crying over.People that talk about you are the ones that are jeaslou dont let that bother you, if your friend talks about you than she aint your friend.And if you really have a good heart than show it find better people to hang out with thats all i have to say.
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Ask angel201 A Question
answered Sunday April 29 2007, 7:58 pm:
Well first of all.
Yes you may be a great person, but that never gives you a right to point fingers, (because there are 3 pointing back at you) or call anyone a hypocrite, people who call people hypocrites are usually hypocrites themself. Honestly, life is too short to be angry, hold grudges, or waste time on drama like this. Either talk it out with them, or stop being friends with them. I know your cousin is your cousin, and you're family, but that doesnt mean you have to be friends. If things cant be fixed, move on with life. Obviously they dont care enough about you, if they are talking about you.
Try making new friends, I realize that is WAY easier said than done. But trust me, you dont want to waste your life away over drama, and calling people hypocrites for things that upset you.
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answered Sunday April 29 2007, 7:56 pm:
Sounds like they are jealous of you.
People that talk badly about other people are usually just jealous.
Let it slide off your back. It isn't worth the tears or fret.
They might become best friends because they both are gossipers, inseure of themselves and like talking crap about other people to make themselves feel better about themselves.
You are better than both of them.
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NEW ORLEANS (AP) One guy is a bruiser of a back, just as comfortable running through defenders as around them.
SAN DIEGO (AP) Tiger Woods is a winner again at Torrey Pines, and the only question Monday was how long it would take him to finish.
Super Bowl week is beginning to resemble one of those family reunions where your crazy uncle says something outrageous, but just true enough to spark a discussion worth having.
NEW ORLEANS (AP) The Baltimore Ravens' reputation for trash-talking arrived at the Super Bowl before they did.
NEW ORLEANS (AP) As Terrell Suggs made his way to his seat for his first media appearance of Super Bowl week, a member of the Baltimore Ravens' PR staff pointed at the linebacker and whispered to


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text: On 11 May 2004, skaller wrote: > On Tue, 2004-05-11 at 17:36, Nicolas Cannasse wrote: > > > > Is there a particular reason that a new data type was invented? Is it > > > still possible to switch to streams? > > > Enum permit also the cloning of an enumeration at any > > time, > > This isn't quite correct. IF the enum represents a forward > iterator, THEN the enum is clonable, otherwise it isn't. A side note: alterative implementations don't necessarily have this restriction. The OO iterator code I posted a couple of days ago doesn't have this restriction. I've been spending time working in C++ again recently. And I'm returning to my old philosophy of simply assuming if C++ does something, it's wrong. In this specific case, I don't see the advantage of having 18 gazillion different types of iterators/enums. You very rapidly need to know what data structure the iterator came from- at which point the prime benefit of iterators (genericity) has vanished. As an example, I *can* implement a "back" function on Ocaml's singly linked lists: let rec back orig_list curr_loc = match orig_list, curr_loc with | car :: ((foo :: _) as cdr), x :: _ -> if foo == x then orig_list else back cdr curr_loc | _ -> assert false ;; Only problem is- that function is O(n). Opps! So we don't provide it. But at what point is a function to expensive to provide? What if the back function is O(log n) cost- do you provide it or not? What if it's an expensive O(1) cost (lots of lseeks and small reads to find where the last line began, for example)? No. Keep things simple. If you need to do something more complicated with the data, either put it in a more advanced data structure, or require it already be in one (by taking an argument with a type of that data structure, not a enum). Personally, I think I'd ditch the count function before I'd introduce a second class of iterators. -- "Usenet is like a herd of performing elephants with diarrhea -- massive, difficult to redirect, awe-inspiring, entertaining, and a source of mind-boggling amounts of excrement when you least expect it." - Gene Spafford Brian


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City Hall - 208 N. 19th St. - Mattoon, Illinois 61938
The rate for our residents thru our Electrical Aggregation Program is 4.48¢/kW*hr.
Ameren Illinois has announced their electric supply rates for the next 12 months.
They are 5.974¢/kW*hr (June 2015 thru Sept 2015) and 6.219¢/kW*hr (Oct 2015 thru May 2016).
The City of Mattoon, Public Works Department, will begin flushing fire hydrants Monday, May 18, 2015 and continue until all lines are flushed.
Flushing will be conducted weekdays between 7:00 a.m. and 3:00 p.m.
The proposed Fiscal Year 2016 budget for the City of Mattoon is now available for review. The document is marked as "tentative" as it is not yet finalized or approved by the City Council.
Click for weather forecast
City Hall - 208 N. 19th St. - Mattoon, Illinois 61938Open Monday-Friday 8:00 a.m.-4:30 p.m.
Copyright © 2015 City of Mattoon. All Rights Reserved.


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Gingrich Calls For Panel To Look At Rules For In Vitro Clinics
Republican presidential hopeful Newt Gingrich is changing another of his positions in an effort to woo socially conservative voters.
Over the weekend he told churchgoers in Florida that as president he'd work to ban research using stem cells derived from human embryos.
Gingrich has long been a strong backer of federal funding for scientific research. In 2001 his support extended to research on stem cells derived from human embryos left over from in vitro fertilization efforts.
But apparently that's no longer the case.
Now he says he'd ban such research, which, he says amounts to, "the use of science to desensitize society over the killing of babies."
Gingrich is also questioning whether those embryos should be left over in fertility clinics in the first place.
"If you have in vitro fertilization you are creating life. And therefore we should look seriously at what should the rules be for clinics that do that because they're creating life," Gingrich said outside the Exciting Idlewild Baptist Church where he attended services Sunday.
Gingrich opposes abortion and says life begins at conception.
Currently the only federal regulations require such clinics to report their success rates.


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Self-editing skills for novelists
- Film, Media & Entertainment
- Kings Place 90 York Way London N1 9GU, GB
Performance and promotion skills for writers
Guardian Masterclasses
Sunday, May 25, 2014 from 10:00 AM to 5:00 PM (BST)
Take your writing from the page to the stage – and into the viral ether – with confidence
Writing is usually a solitary pursuit. Publishing and promoting a finished work, on the other hand, requires a somewhat more sociable approach. Literary festivals, salons and readings are a great way for new writers to attract the attention of agents and publishers, while book tours and a strong social media presence are promotional mainstays for established authors.
If reading out your work to a room full of strangers sounds like the stuff of particularly visceral nightmares, this course is for you. Novelist Paul Burston has conquered his own stage fright to found and host the Polari Literary Salon at London's Southbank, and has been recognised as one of the top 50 tweeters in the UK. Over the course of this lively, inspiring day, he can show you simple tricks to overcome nerves, inject passion into your readings, and even use the experience to make yourself a better writer. He also offers invaluable tips on building a reputation via social media, another crucial skill in engaging your work with a broader community.
This informal, small-scale event demonstrates practical, easy-to-apply performance and social media strategy tips designed specifically to help authors promote their work to best effect. Topics covered on the day include:
PLEASE NOTE: During the course, Paul will offer one attendee the opportunity to perform a five-minute reading at Polari on Tuesday 27 May. If you are interested in this opportunity, please bear in mind that Polari has a primarily LGBT audience and readings on gender and sexuality issues are best received.
Paul Burston is an author and journalist whose books include the critically-acclaimed novels Shameless (2001), Star People (2006), Lovers & Losers (2007) and The Gay Divorcee (2009). Paul is the host of award-winning LGBT literary salon Polari at the Southbank Centre, and founder of The Polari First Book Prize. Paul also writes for a wide range of publications including the Guardian, the Times and the Independent.
Full course and returns information on the Masterclasses website
Welcome to Guardian masterclasses – a unique programme of learning embedded within one of the world's most forward-thinking media organisations.
Masterclasses offer a broad range of short and long courses across a variety of disciplines from creative writing, journalism, photography and design, film and digital media, music and cultural appreciation.
Harnessing the expertise and specialisms within the organisation, our courses are led by first class and award winning guardian professionals whilst also drawing on the skills and expertise of other leading figures at the forefront of the creative and digital industries.
The programme is aimed at anyone interested in personal or professional development whether that be refining your skills, focusing your ambition or simply broadening your mind and gaining inspiration.
In order to purchase these tickets in installments, you'll need an Eventbrite account. Log in or sign up for a free account to continue.


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You can now invite anyone to meet at your favorite Starbucks.
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42 year old man | Brooklyn, NY, USA | Seeking women 27-35 within 10 miles
Active within 24 hours
I have learned a few things over the years...
I have learned a few things over the years what important in a good solid relationship and what I want, I would like someone who can hold a good conversation, some one with a good sence of humor,. Some one that likes to have fun even if we are just taking a walk in a park, A person who's is independent. Some one who can appreciate flowers just because it's Tuesday. Some who is career orentated but enjoys her down time. And........... A strong foundation starting with friendship I am looking that person who is truly looking for a future. If you have a good sence of humor, hard working, easy going, laid back maybe you can be my queen.
Cooking , Dining out , Hobbies and crafts , Movies/Videos , Music and concerts , Playing sports , Watching sports
Billiards / Pool, Bowling, Running, Walking / Hiking
I exercise 1-2 times per week.
I like Cats, Dogs, Exotic Pets, Horses, Reptiles
Middle of the Road
Sports,Traveling, Dining
Museums, cooking and spending time with family
World War Z
Z{{Cta}} {{#if HasMoreInfo}} ?{{CtaSecondary}} {{/if}}
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|Social Drinker
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Warsaw is a business city first and foremost, and occupancy rates reflect that. Prices dip the moment the clock hits Friday, 5pm, and you'll find some great discounts available if you hunt around online. A good place to start is at poland.inyourpocket.com where our Hotel Calculator scours booking engines for the best rates based on your criteria (you can thank us later). The Warsaw hotel market reflects the city's image as the corporate briefcase of Eastern Europe and comes well equipped with five star offerings as well as a new breed of options for thrifty travellers. Warsaw now has a group of Golden Keys concierges and their tips for the coming months can be found in our Ask the Concierge box.
Here is a list of recommendations depending on what you are looking for.


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Starring: Tracy Neff* as Eliza Doolittle, Warren Kimmel* as Henry Higgins, Thomas Lamont as Freddy Eynsford-Hill, John Payne* as Alfred Doolittle, Barbara Pollard* as Mrs. Higgins with the Ensemble: Juilia Bazso, Suzanne Bazso, Paul Beckett, Jonathan Boudin, Lyndsey Britten, Geoff Campbell, Daniel Cardoso, Emily Cawley, Amy Gartner, Christopher Hall, Tiffany Hambrook, Justin Lapena, Kyle Oliver, Kathryn Palmer, Maddison Popov, Matt Ramer, James Rha, Cassady Ranford, Michael Seeley, Peter Stainton, Stefanie Swinnard, Julia Ullrich, Krista Wallace, Claire Wardle, Michael Wild, Chelsey Yamasaki
(on Wednesday April 22nd, the role of Eliza Doolittle will be played by Emily Cawley)
Artistic Director Valerie Easton
Director Max Reimer *
Musical Director James Bryson
Choreographer Suzanne Ouellette *
Set Designer Brian Ball
Costume Designer Christina Sinosich
Lighting Designer Gerald King
Sound Designer Tim Lang
Associate Director Cory Haas
Technical Director Mike Cooper
Head of Props Jennifer Stewart
Stage Manager Ingrid Turk *
Assistant Stage Manager Courtney Bettanin
Assistant Stage Manager Gerri Torres
* These Artists appear with the permission of Canadian Actors' Equity Association.
Become a member of RCMT for $10, save on all tickets, access tickets one week earlier than the launch date!
Share Your Passion for Great Musical Theatre
If you believe, as we do, that musical theatre enriches the lives of those on the stage, behind the scenes and in our audience. Then we ask you to join us and help us on our journey from New York City to London for our 2015 production of MY FAIR LADY!
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Please make a tax-deductable gift to RCMT today and take a lead role in creating great theatre.
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It directly supports our annual production – helping to cover the cost of equipment and supplies to build sets, props and costumes, and to maintain the high quality musical theatre for which RCMT is known. You also support apprenticeship and training opportunities for emerging theatre artists. That you also touch the lives of many young people with big dreams – that's a gift that reaches beyond RCMT.
Please give today by calling 604-521-505 or send your donation to Royal City Musical Theatre c/o the Massey Theatre at 735 8th Ave New Westminster BC V3M 2R2. Your support is invaluable and gratefully received.
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We are creating a new personal empowerment web show that will feature topics that include bully-proofing, confidence-building, self defense, etc. Looking for a short, dynamic opening for the show with title, credits, music etc. The show will be more like a video blog post about 10-15 min long.
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14 Beloved Children's Characters Turned "Sexy" Halloween Costumes
1. The Lorax
2. Big Bird
3. Strawberry Shortcake
This is billed as a "Sexy Strawberry Dessert Costume," but the source is clear.
4. My Little Pony
5. The Grinch
6. Rainbow Brite
This one's called "Sexy 80s Rainbow Vixen."
7. Cheshire Cat
8. Cookie Monster
h/t Jessica Valenti.
9. Tweety Bird
10. Ninja Turtle
11. Power Ranger
12. Ursula the Sea Witch
13. Glinda the Good Witch
14. Kermit the Frog
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- The U.S. and Cuba have agreed to open embassies in Washington and Havana as they restore diplomatic ties after more than 50 years.
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Scotch Crunchies
21 Dec 2010
|Butterscotch pieces
|6 Ounce (1 Package)
|Semisweet chocolate/Imitation chocolate-flavored pieces
|6 Ounce, flavored
|Canned chow mein noodles
|3 Ounce (1 Can)
|1 Cup (16 tbs)
In large glass bowl place butterscotch and chocolate pieces.
Micro-cook, uncovered, till melted, about 2 1/2 minutes; stir after each minute.
Stir in noodles and marshmallows or raisins.
Drop by teaspoon onto waxed paper.
Chill, if desired.
Recipe Summary
Difficulty Level:Easy
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Scotch Crunchies Recipe


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"If you want to see the world, you better get going" feats's Profile
I always wanted a picture of the New York skyline. I was in Manhattan chasing after a visa to visit Ukraine late summer 2001 when I thought about heading over to Brooklyn to look west over the river for the view.
Meanwhile, the Ukraine people have no concept of what a line is and bring that mentality to their Consulate office here in the states. Here, we are trained like rodents to follow that rat-maze to the counter that we have at official offices, banks, carnival rides, etc. For the Uke's (Russians - and some other peoples of the world for that matter) it's a landrush! Whoever is serious about getting it done, is up there pushing and shoving for position.
Anyway I was tired from the visa process and was anxious to get out of the city and started on the 4-hour drive home.
"I'll get that picture some other time." At least that's what I thought. "It's the New York skyline, it'll always be there!" Instead, I decided to give it a go and at least try to get to a place for a view.
I felt so sick seeing those towers come down and what it did to so many people. On some smaller level, I felt a kind of connection having been there several weeks before. One thought I had is written as the title to this page.
This place is so beautiful.
Every two feet you can take a photograph and look like a professional. You can close your eyes, trip and accidently take great photos all over the place. When I grow old and go blind, this is where I'll go to take pictures...
The beautiful waters lure you in, then the warm white sands calm you to sleep. Painfully, you wake up to the sunburn from hell. This is where you go when you're in the mood to peel layers of skin off.
These beautiful churches have colorful frescoes depicting religious history typically for those who could not read. This country has a great "campers" mentality. Wherever you are, whenever you're in the mood, just pitch a tent. You know, pick up your mess when you're done; nobody cares...
This desert island majors in volcanic rock and spiny cactuses. Its reputation as a tropical paradise disguise its dry as death geographic interior. This sole twisted tree is actually quite famous on the island and draws people for many purposes. You'll see those who picnic all over it, some getting married next to it, screaming kids hanging from it, people buttin' out their cigarettes on it, urinating dogs tied to it, etc. It's a shame; it's lived a long and difficult life. I was able to get this picture before these people tied a rope between it and their car to dry their clothes.
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Comments (7)
I love the view of the NYC skyline from the Brooklyn Bridge. Great shot! ~
Greetings From Syr. I hope you're someplace warmer now.
How is Chile?! I'm dying to go! What is your # thing for someone to do when they are there?
Very nice page! The pics are beautiful! Greetings from Fortaleza! =)
I agree with you even New Yorkers who live and work in the heart of Manhattan don't take true advantage of our great sites. I don't take the NYC skyline for granted anymore! Thanks for your picture of WTC..
you're pretty well travelled! Greetings from close by!
Interesting summary impressions that are also well illustrated! Thanks.
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Les Belles Amour
Les Belles Amour
by Nina Ricci
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- Design House: Nina Ricci
- Fragrance Notes: a sweet, green floral with mint, sweet woods and sugar.
- Year Introduced: 1999
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"Lovely villa with beautiful sunset views"
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Be Careem: 6 Ramadan initiatives the car hire app is supporting
How will you make a difference this Holy Month?July 2, 2015 3:00
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Abu Dhabi's TAQA drops 5.4 pct after saying no merger talks
Qatar's Doha Bank raises 2 bln riyal Tier 1 capital-boosting bond
Dubai's Drake & Scull increases GCC ownership limit to 100 pct
Abu Dhabi's TAQA says not holding merger talks
Gold languishes below $1,170 ahead of U.S. jobs data
Kuwait parliament approves budget with $27 billion deficit
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European tour operators evacuate thousands from Tunisia after attack
Dr. Soliman Fakeeh Hospital (DSFH), one of the leading names in the regional healthcare sector, today announced that it has unveiled the first-of-its-kind healthcare Corporate Social Responsibility (CSR) Report in the region, aimed at reinforcing its commitment to sustainability and responsibility to the social environment it operates in.
January 27, 2010 12:00 by Aarti Nagraj
Dr. Soliman Fakeeh Hospital unveils first-of-its-kind healthcare CSR report in the
Unique Sustainability Report covers patient care, workplace
relationships, community wellbeing and environment
Dr. Soliman Fakeeh Hospital (DSFH), one of the leading names in the regional healthcare sector, today announced that it has unveiled the first-of-its-kind healthcare Corporate Social Responsibility (CSR) Report in the region, aimed at reinforcing its commitment to sustainability and responsibility to the social environment it operates in.
The DSFH report is also the first non-financial report to be developed by a hospital in the region. The CSR report not only looks at steps to raise standards of patient treatment and care but also reinforces DSFH's responsibility towards the local community in order to improve the quality of life and productivity.
As per the report, DSFH has implemented a series of initiatives such as cutting down on energy and water consumption, switching to eco-friendly technology and programs aimed at cutting down on its carbon footprint. The report, developed with international standards, was applauded and approved by Global Reporting Initiative (GRI).
In his comments on the extraordinary report, Dr. Mazen Fakeeh, General Director at DSFH, said: "Dr. Soliman Fakeeh Hospital has always believed in practices built around the principles of honesty, responsibility and ethics. Over the last few years, we have established our mandate as a socially responsible hospital by cutting down on energy and water use, switched to digital x-rays, started recycling campaign, and tracking our carbon emissions. Our major objective for the next years is to raise the patient satisfaction rate by at least five per cent annually. These were the first steps towards a cleaner, greener future for
He added: "As part of the steps in this direction, we have fixed nearly 540 water saving installations following recommendations by the Jeddah Municipality; which helped reduce our water consumption in 2008 by more than 50 per cent compared to the previous year. We also put in place a number of energy saving measures which have reduced our energy consumption by 5 per cent compared to the previous years. Our efforts are primarily aimed at embedding sustainability into every aspect of our strategy and decision-making -- on patient care, workplace relationships, community wellbeing and environment."
The company's moves, highlighted in the CSR Report, centered on a philosophy that sustainable businesses can cut costs, drive innovation, reduce risk and motivate employees. Apart from this, DSFH also ensured safe and healthy working conditions for our staff by adopting the Occupational Safety and Health Administration (OSHA) and National Institute of Occupational Safety and Health (NIOSH) standards for prevention of work-related illness and injuries.
The Arab Health Congress 2010 was attended by a large number of healthcare professionals and representatives from around the region.
About Dr Soliman Fakeeh Hospital
Dr Soliman Fakeeh Hospital (DSFH), founded in Jeddah in 1978, is a leading referral hospital. The hospital is committed to the provision of preventive and therapeutic health-care in all medical and surgical specialties. DSFH has been a true leader in the field, whose pioneering spirit and visionary resolve has – for over three decades – advanced by leaps the standards of healthcare delivery in the Kingdom and in the region. In 1986, the first expansion of the Hospital was inaugurated by His Majesty, the late King Fahd bin Abd al-Aziz. In addition to doubling the facility's inpatient capacity, that expansion introduced such new centers as open-heart surgery and new clinics, including neurosurgery, neurology, nephrology and infertility clinics, and established DSFH as a definitive leader of private healthcare in the Kingdom. In 1999, a second expansion inaugurated by the Custodian of the Two Holy Mosques King Abdullah bin Abd Al-Aziz, Crown Prince at the time, marked the addition of two new structures to the DSFH campus.
DSFH is a regional pioneer in the field of organ-transplantation, including kidney, bone-marrow, liver, and heart transplantation. Its open-heart surgery center claims the highest number of operations done in the private sector in the Kingdom with a success rate comparable to any international center of excellence. DSFH was the first private hospital in the Western Region of the Kingdom to become accredited by the Joint Commission International (JCI) in 2006, and by the Australian Council for Healthcare Standards International (ACHSI) in 2008. DSFH is presently considered one of the most distinguished hospitals in the
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