The first step in the analysis of these data is to test if there is a significant relationship between air temperature and oxygen consumption for each species in other words test Ho: β = 0. Inspection of the figure indicates that: a) there appears to be a negative relationship between oxygen consumption and ambient air temperature in both species, and b) the relationships appear very similar for the two species. To illustrate how ANCOVA works, let’s consider the relationship of oxygen consumption to air temperature in two bird species that co-occur in semi-desert areas of the southwest (Table 16.1, Figure 16.1). The null hypothesis is simply that the elevations are equal (the lines coincide) while the alternative hypothesis is that the elevations are not equal (the lines do not coincide). Furthermore, ANCOVA also allows for testing whether the elevations are equal for those regression lines, which are demonstrated to come from populations with equal slopes. ANCOVA can also be used to compare only two lines where Ho: β1 = β2 and Ha: β1 ≠ β2.

It allows for testing Ho: β1 = β2 = β3…= βk with an alternative hypothesis that the regression lines were not derived from samples estimating populations which all had equal slopes. The procedure, which must be used in comparing more than two lines, is known as an analysis of covariance (ANCOVA). However, t-tests cannot be used when there are more than two lines thus, t-tests are limited in their application and their use is not covered in this course.

In cases where there are only two lines, a t-test can be used to test the difference between slopes, and if the slopes are equal, then another t-test can be used to determine if the difference in elevation is significant. The test for coincidence (elevation, or same y-intercept) can only be carried out for those lines that have been demonstrated to have no significant statistical difference in slope. If it is concluded that the regression lines are parallel, that is that the slopes are equal, then there is often interest in determining whether the regressions have the same elevation (i.e.

In these cases, the interest is in whether the sample slopes (b) are estimates of the same or different population slopes (β). Multiple Regression Analysis (ANCOVA) In many cases biologists are interested in comparing regression equations of two or more sets of regression data.

Excel linear regression formula 16.6