in the full model matrix is +*pd this creates an increasing sequence of "nestedmodel matricesx [ ][ xd ]from (saythe baseline normal model matrix to the full model matrix think of each model matrix corresponding to specific variables in the model we follow similar projection procedure as in figure first project onto span(xto yield the vector (dthen project (donto span([ xd- ]to obtain ( - and so onuntil ( is projected onto span( to yield ( (in the case that by applying pythagorastheoremthe total sum of squares can be decomposed as ky ( ky (dk ky (dy ( - ky ( ( { { { { df= - df= - ( df= df=pd software packages typically report the sums of squares as well as the corresponding degrees of freedom (df) ppd degrees of freedom coefficient of determination to assess how linear model xb compares to the default model ewe can compare the variance of the original dataestimated via (yi ) / ky /np yi ) / kb with the variance of the fitted dataestimated via ( /nwhere xb the sum (yi / ky is sometimes called the total sum of squares (tss)and the quantity kb ( ky is called the coefficient of determination of the linear model in the notation of figure ( and ( so that total sum of squares coefficient of determination ky ( ( ky ( ky ( tss rss tss ky ( ky ( note that lies between and an value close to indicates that large proportion of the variance in the data has been explained by the model many software packages also give the adjusted coefficient of determinationor simply the adjusted defined by adjusted ( - np the regular is always non-decreasing in the number of parameters (see exercise )but this may not indicate better predictive power the adjusted compensates for this increase by decreasing the regular as the number of variables increases this heuristic adjustment can make it easier to compare the quality of two competing models adjusted coefficient of determination