PSY 555 Homework 19

1.              ANOVAs and regression analyses are not really different procedures.  In fact, ANOVA is a special case of regression.  Both analyses utilize the F-distribution.  ANOVA and regression examine similar factors in the data (e.g., variances and standard errors) and synthesize the relations between these factors in a statistical result.  ANOVAs are calculated using a mathematical simplification of regression analyses and are less powerful than regression analyses.  Thus, ANOVAs are generally conducted when a regression is not appropriate (e.g., when portions along the continuum of the X independent/predictor variable have not been observed and assessed).  In the past, ANOVAs were commonly used because regression analyses are complex mathematically and so they were difficult to calculate by hand (this factor was a weakness of regression analyses in the past—this is not an issue anymore because of statistical computer programs that complete these calculations easily).  Both are examples of a general linear model.  The difference between ANOVA and regression analyses is largely a matter of what can be concluded from the results of these analyses.  ANOVAs determine whether different groups, or treatments, have different means, while multiple regressions tell us how and if a mean is related to the treatments of inclusions (of predictors) in a group.

2.              A t-test compares the means of two groups in order to determine if the means are significantly different.  With only two groups, we can easily conclude, if the means are different, which group has higher or lower scores on a measurement.  An ANOVA also allows us to compare means and determine whether there are significant differences among them, but is conducted when there are more than two means (groups/treatments; a t-test between independent groups is actually an ANOVA).  ANOVAs also allow for using more than one independent variable to be examined and determine the effect of individual variables, as well as interactions among variables.

3.              The assumptions underlying regression are similar to those of regression.  These assumptions include homogeneity of variance, normality of the distributions, and the independence of observations.  The homogeneity of variance assumption refers to the fact that ANOVA assumes each of our samples (treatments/conditions) has approximately the same variance.  The normality assumption refers to the fact that we assume the dependent variable (in the book the example was recall scores) is normally distributed for each of our treatments/conditions.  Finally, in ANOVAs, we assume that observations are independent of one another—that is, that is we know how one observation in an experimental condition stands relative to that condition’s mean, it tells us nothing about the other observation.

4.              The three sources of variation for simple ANOVAs are the total variation, the error variation, and the treatment variation.  The total variation (SSTOT) allows you to determine how much the dependent variable observations deviate from the dependent variable mean (across all observations)—essentially the amount of variation in the dependent variable.  The error variation, SSERR, reflects the amount of variation that treatments do not explain.  The treatment variation, SSTREAT, tells us how much total variation treatments account for.

5.

N=50 per group

 21 -3.4 11.56 23 -1.4 1.96 27 2.6 6.76 22 -2.4 5.76 29 4.6 21.16

N=5       2

N=5

S=

The best estimate of the population variance is 591.68.