title: Adding vs. removing interaction changes significance of single predictors in lm()
body: <p>I'm running two types of ANCOVA's in R on the same data. One model includes the interaction term between the two predictors, the other doesn't:</p>
  <pre><code>model&lt;-lm(trait~size*population, data=d)
  model2&lt;-lm(trait~size+population, data=d)
  </code></pre>
  <p>When looking at the t-statistics of the first model using summary(model), the interaction term appears non-significant and highly trivial (t-value is -0.14). Similarly, there seems to be no effect of &quot;population&quot; (t = -0.13, p-value = 0.897). However, summary(model2) indicates a quite strong effect of &quot;population&quot; on &quot;trait&quot; (t = -3.3, p-value = 0.0016). How is this possible?</p>
  <p>Here are the data:</p>
  <pre><code>structure(list(ID = c(&quot;rat1&quot;, &quot;rat2&quot;, &quot;rat3&quot;, &quot;rat4&quot;, &quot;rat5&quot;, 
  &quot;rat6&quot;, &quot;rat7&quot;, &quot;rat8&quot;, &quot;rat9&quot;, &quot;rat10&quot;, &quot;rat11&quot;, &quot;rat12&quot;, &quot;rat13&quot;, 
  &quot;rat14&quot;, &quot;rat15&quot;, &quot;rat16&quot;, &quot;rat17&quot;, &quot;rat18&quot;, &quot;rat19&quot;, &quot;rat20&quot;, 
  &quot;rat21&quot;, &quot;rat22&quot;, &quot;rat23&quot;, &quot;rat24&quot;, &quot;rat25&quot;, &quot;rat26&quot;, &quot;rat27&quot;, 
  &quot;rat28&quot;, &quot;rat29&quot;, &quot;rat30&quot;, &quot;rat31&quot;, &quot;rat32&quot;, &quot;rat33&quot;, &quot;rat34&quot;, 
  &quot;rat35&quot;, &quot;rat36&quot;, &quot;rat37&quot;, &quot;rat38&quot;, &quot;rat39&quot;, &quot;rat40&quot;, &quot;rat41&quot;, 
  &quot;rat42&quot;, &quot;rat43&quot;, &quot;rat44&quot;, &quot;rat45&quot;, &quot;rat46&quot;, &quot;rat47&quot;, &quot;rat48&quot;, 
  &quot;rat49&quot;, &quot;rat50&quot;, &quot;rat51&quot;, &quot;rat52&quot;, &quot;rat53&quot;, &quot;rat54&quot;, &quot;rat55&quot;, 
  &quot;rat56&quot;, &quot;rat57&quot;, &quot;rat58&quot;, &quot;rat59&quot;, &quot;rat60&quot;, &quot;rat61&quot;, &quot;rat62&quot;, 
  &quot;rat63&quot;, &quot;rat64&quot;, &quot;rat65&quot;, &quot;rat66&quot;, &quot;rat67&quot;, &quot;rat68&quot;), population = c(&quot;A&quot;, 
  &quot;A&quot;, &quot;A&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, 
  &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, 
  &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, 
  &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, &quot;A&quot;, 
  &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, &quot;B&quot;, 
  &quot;B&quot;, &quot;B&quot;), size = c(39.72, 46.72, 38.37, 40.8, 46.57, 35.93, 
  51.69, 40.97, 45.39, 43.67, 43.68, 39.2, 45.07, 42.11, 46.91, 
  45.99, 42.43, 41.36, 42.54, 38.41, 42.35, 40.79, 45.32, 43.67, 
  46.34, 40.26, 39.09, 49.2, 47.85, 45.14, 42.38, 44.2, 41.22, 
  41.52, 45.12, 45.63, 44.15, 40.18, 47.88, 43.86, 42.79, 43.99, 
  47.22, 43.78, 41.84, 42.99, 37.9, 44.21, 47.82, 45.26, 45.97, 
  44.99, 40.48, 42.27, 43.85, 41.1, 42.11, 41.9, 38.17, 46.08, 
  37.75, 40.92, 38.69, 46.34, 39.3, 49.76, 43.69, 37.18), trait = c(3.657, 
  4.096, 3.186, 4.286, 3.901, 2.882, 4.666, 4.635, 4.93, 4.264, 
  4.329, 3.493, 5.142, 4.859, 4.272, 3.5, 4.199, 4.434, 4.278, 
  3.211, 4.382, 3.941, 4.525, 4.547, 4.08, 4.345, 3.827, 4.822, 
  4.363, 4.229, 4.063, 4.605, 3.803, 4.008, 4.775, 3.949, 4.308, 
  4.048, 4.697, 3.951, 3.488, 3.705, 3.408, 4.458, 3.834, 4.057, 
  3.318, 4.04, 4.596, 4.931, 4.294, 3.817, 4.16, 4.304, 3.67, 4.273, 
  4.194, 3.461, 3.072, 3.692, 3.019, 3.17, 3.095, 3.785, 3.814, 
  4.131, 3.639, 3.385)), class = &quot;data.frame&quot;, row.names = c(NA, 
  -68L))
  </code></pre>