LECTURE 15 Hypotheses about Contrasts EPSY 640 Texas A&M University

Hypotheses about Contrasts C = c 1  1 + c 2  2 + c 3  3 + …+ c k  k, with  c i = 0. The null hypothesis is H 0 : C = 0 H 1 : C  0

Hypotheses about Contrasts C 1 = (0)  instruction + (1)  advance organizer (-1)  neutral topic Thus, for this contrast we ignore the straight instruction condition, as evidenced by its weight of 0, and subtract the mean of the neutral topic condition from the mean for the advance organizer condition. A second contrast might be 2, -1, -1: C 2 = (2)  instruction (-1)  advance organizer (-1)  neutral topic We can interpret this contrast better by examining its null hypothesis: C 2 = 0 = (2)  instruction (-1)  advance organizer (-1)  neutral topic, so that (2)  instruction = (1)  advance organizer + (1)  neutral topic and  instruction -[ (1)[  advance organizer + (1)  neutral topic ] / 2 = 0.

Contrasts simple contrasts, if only two groups have nonzero coefficients, and complex contrasts for those involving three or more groups

Planned Orthogonal Contrasts Orthogonal contrasts have the property that they are mathematically independent of each other. That is, there is no information in one that tells us anything about the other. This is created mathematically by requiring that for each pair of contrasts in the set,  c i1 c i2 = 0, where c i1 is the contrast value for group i in contrast 1, c i2 the contrast value for the same group in contrast 2. For example, with C 1 and C 2 above, C 1 :01-1 C 2 : C 1 C 2 :0 x 2 +1 x –1 +-1 x –1 = 0 –1 + 1 = 0

Planned Orthogonal Contrasts VENN DIAGRAM REPRESENTATION SSy Treat SS SSc1 SSerror R 2 c1=SSc1/SSy SSc2 R 2 c2=SSc2/SSy R 2 y=(SSc1+SSc2)/SSy

Geometry of POCs C1: 0, 1, -1 C2: 2, -1, -1 GP 1 GP 2 GP 3

PATH DIAGRAM FOR PLANNED ORTHOGONAL CONTRASTS C2C2 e C1C1  1 (r c1,y )=.085  2 (r c2,y ) =.048 y

Nonorthogonal Contrasts VENN DIAGRAM REPRESENTATION SSy Treat SS SSc2 SSerror SSc1

PATH DIAGRAM FOR PLANNED NONORTHOGONAL CONTRASTS C2C2 e C1C1  1 (r c1,y )=.128  2 (r c2,y ) = y r=.78

Control Treatment Treatment+Drug Treatment+ Placebo C T TD TP The purpose of the placebo is to mimic the results of the drug. An even more complex design might include a control plus the placebo. The set of orthogonal contrasts follow from hypotheses of interest: C T TD TP C 1 : This contrast assesses whether treatments are more effective generally than the control condition.

Control Treatment Treatment+Drug Treatment+ Placebo C T TD TP C 2 : This contrast compares the treatment with additions to treatment. C 3 : and this contrast compares the effect of the drug with the placebo. There are other sets of contrasts a researcher might substitute or add. Here, we will look at the contrasts to determine that they are orthogonal: C 1 : C = 0, so that C 1 and C 2 are orthogonal.

Control Treatment Treatment+Drug Treatment+ Placebo C T TD TP C 1 : C = 0, so that C 1 and C 3 are orthogonal. C 3 : C –1 +1 = 0, so that C 3 and C 2 are orthogonal.

A second set of contrasts might be developed as follows: C T TD TP C 1 : This contrasts the control with the primary drug conditions of interest. Next, C 2 : This contrast compares the treatment with treatment plus drug, the major interest of the study. Finally C 3 : and this contrast compares the effect of the drug with the placebo. C 1 : C = 0, so that C 1 and C 2 are orthogonal. C 1 : C = -1, so that C 1 and C 3 are not orthogonal. C 3 : C –1 0 = -1, so that C 3 and C 2 are not orthogonal.

Polynomial Trend Contrasts When groups represent interval data we can conduct polynomial trend contrasts example: Group A receives no treatment, Group B 10 hours, and group C receives 20 hours of instructional treatment Treatment condition (time) is now interval:

Polynomial Trend Contrasts The contrast coefficients for polynomial trends fit curves: linear, quadratic, cubic, etc. The coefficients can be obtained from statistics texts most easily SPSS has a polynomial trend option in the Analyze/Compare Means/One Way ANOVA analysis

Polynomial Trend Contrasts: example of drug dosages ml dose C 1 : linear C 2 : quadratic C 3 : cubic

C1C1 C2C C3C3 Fig. Graphs of planned orthogonal contrasts for four interval treatments

SPSS EXAMPLE The groups represent the five quintiles of school enrollment size, 1-281, , , , and

SPSS EXAMPLE Unweighted used because each group has the same # of schools

SPSS EXAMPLE We might have gotten a quadratic from this curve but too much variation within groups