Psychology 202a Advanced Psychological Statistics November 24, 2015

The plan for today A priori contrasts in SAS Orthogonal contrasts Contrast coding The Eysenck ANOVA example Helmert contrasts Introduction to power

A priori contrasts A contrast is a question about a linear combination of means. Example: Shorthand notation: 1/2 1/2 -1 Equivalent: Another question that might interest us is

Contrasts (continued) Once a contrast is specified, its sum of squares is calculated: Contrasts always have 1 df, so the sum of squares is a mean square. Division by the error mean square provides an F statistic that tests the contrast.

Contrasts (continued) Illustration in SAS. Any set of contrasts defined in advance may be tested, dividing the alpha among them. However, this particular set has a special property: orthogonality. If the contrasts are orthogonal and specified in advance, there is no need for an adjustment to alpha.

Checking for orthogonality Multiply the corresponding coefficients of each pair of contrasts. If the products sum to zero, the pair is orthogonal. Here, we are considering (1, 1, -2) and (1, -1, 0). (1×1) + (1×-1) + (-2×0) = 0, so the pair is orthogonal.

Why is orthogonality special? Contrast coding Illustration in SAS So orthogonal contrasts divide the model sum of squares into exhaustive and mutually exclusive partitions. A more complicated example (Eysenck memory experiment)

Introducing power In the world of hypothesis testing, one of two things is true: –The null hypothesis may be true; or –The null hypothesis may be false. In the world of hypothesis testing, one of two outcomes will occur: –The null hypothesis may be rejected; or –The null hypothesis may be retained.

Consider that in tabular form: H 0 TrueH 0 False H 0 Rejected H 0 Retained

Consider that in tabular form: H 0 TrueH 0 False H 0 RejectedGreat! H 0 Retained

Consider that in tabular form: H 0 TrueH 0 False H 0 RejectedGreat! H 0 RetainedNo problem.

Consider that in tabular form: H 0 TrueH 0 False H 0 RejectedType I errorGreat! H 0 RetainedNo problem

Consider that in tabular form: H 0 TrueH 0 False H 0 RejectedType I errorGreat! H 0 RetainedNo problemType II error

Consider that in tabular form: H 0 TrueH 0 False H 0 Rejected Type I error (p =  Great! H 0 RetainedNo problemType II error

Consider that in tabular form: H 0 TrueH 0 False H 0 Rejected Type I error (p =  Great! H 0 RetainedNo problem Type II error (p = 

What is power? In that scenario, power = 1 –  In other words, power is the probability that we will avoid a Type II error, given that the null hypothesis is actually false.