LECTURE 15Hypotheses about Contrasts

EPSY 640

Texas A&M University

Hypotheses about Contrasts

C = c11 + c22 + c33 + …+ ckk , with ci = 0 .

The null hypothesis is

H0: C = 0

H1: C 0

Hypotheses about ContrastsC1 = (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:

C2 = (2)instruction (-1)advance organizer (-1)neutral topic

We can interpret this contrast better by examining its null hypothesis:

C2 = 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,

ci1ci2 = 0,

where ci1 is the contrast value for group i in contrast 1, ci2 the contrast value for the same group in contrast 2. For example, with C1 and C2 above,

C1 : 0 1 -1

C2: 2 -1 -1

C1C2: 0 x 2 + 1 x –1 +-1 x –1

= 0 –1 + 1

= 0

Planned Orthogonal Contrasts• VENN DIAGRAM REPRESENTATION

SSy

Treat SS

SSc1SSerror

R2c1=SSc1/SSy

SSc2

R2c2=SSc2/SSy

R2y=(SSc1+SSc2)/SSy

Geometry of POCs

C1: 0, 1, -1

C2: 2, -1, -1

GP 1

GP 2GP 3

PATH DIAGRAM FOR PLANNED ORTHOGONAL CONTRASTS

C2

e

C1 1 (rc1,y)=.085

2 (rc2,y) = .048

Coefficientsa

50.568 .851 59.398 .000

1.132 1.245 .048 .909 .364

.731 .456 .085 1.603 .110

(Constant)

c2

c1

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: t10a.

ANOVAb

493.868 2 246.934 2.439 .089a

39492.208 390 101.262

39986.076 392

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), c1, c2a.

Dependent Variable: t10b.

y

Nonorthogonal Contrasts• VENN DIAGRAM REPRESENTATION

SSy

Treat SS

SSc2SSerror

SSc1

PATH DIAGRAM FOR PLANNED NONORTHOGONAL CONTRASTS

C2

e

C11 (rc1,y)=.128

2 (rc2,y) = -.022

y

r=.78

ANOVAb

493.868 2 246.934 2.439 .089a

39492.208 390 101.262

39986.076 392

Regression

Residual

Total

Model1

Sum ofSquares df Mean Square F Sig.

Predictors: (Constant), H-W DIFFS, B-W DIFFSa.

Dependent Variable: SELF ESTEEMb.

Coefficientsa

50.568 .851 59.398 .000

1.863 1.178 .128 1.582 .114

-.402 1.460 -.022 -.275 .783

(Constant)

B-W DIFFS

H-W DIFFS

Model1

B Std. Error

UnstandardizedCoefficients

Beta

StandardizedCoefficients

t Sig.

Dependent Variable: SELF ESTEEMa.

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

C1 : 3 -1 -1 -1

This contrast assesses whether treatments are more effective generally than the control condition.

Control Treatment Treatment+Drug Treatment+ Placebo

C T TD TP

C2: 0 2 -1 -1

This contrast compares the treatment with additions to treatment.

C3: 0 0 1 -1

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:

C1: 3 -1 -1 -1

C2 0 2 -1 -1

0+ -2 +1 +1 = 0, so that C1 and C2 are orthogonal.

Control Treatment Treatment+Drug Treatment+ Placebo

C T TD TP

C1: 3 -1 -1 -1

C3 0 0 1 -1

0 + -0 -1+1 = 0, so that C1 and C3 are orthogonal.

C3: 0 0 1 -1

C2 0 2 -1 -1

0 + -0 –1 +1 = 0, so that C3 and C2 are orthogonal.

A second set of contrasts might be developed as follows:

C T TD TP

C1 : 2 -1 -1 0

This contrasts the control with the primary drug conditions of interest. Next,

C2: 0 1 -1 0

This contrast compares the treatment with treatment plus drug, the major interest of the study. Finally

C3: 0 0 1 -1

and this contrast compares the effect of the drug with the placebo.

C1: 2 -1 -1 0

C2 0 1 -1 0

0 +-1 +1+0 = 0, so that C1 and C2 are orthogonal.

C1: 2 -1 -1 0

C3 0 0 1 -1

0+ 0 -1+0 = -1, so that C1 and C3 are not orthogonal.

C3: 0 0 1 -1

C2 0 1 -1 0

0 + -0 –1 0 = -1, so that C3 and C2 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:0 10 20

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

0 100 200 300 ml dose

C1 : -3 -1 1 3 linear

C2: -1 1 1 -1 quadratic

C3: -1 3 -3 1 cubic

3210-1-2-3

0 100 200 300 0 100 200 300

3210-1-2-3

0 100 200 300

C1 C2

3210-1-2-3

C3

Fig. Graphs of planned orthogonal contrasts for four interval treatments

SPSS EXAMPLEDescriptives

g3rall00 Grade 3 Reading TAKS score 2000

455 86.8760 13.98108 .65544 85.5880 88.1641 20.00 100.00

612 87.5147 11.47839 .46399 86.6035 88.4259 33.30 100.00

611 87.1545 10.83670 .43841 86.2935 88.0155 39.10 100.00

608 86.8724 10.46278 .42432 86.0391 87.7057 40.30 100.00

607 86.0428 11.91949 .48380 85.0927 86.9930 27.80 100.00

2893 86.8944 11.67421 .21705 86.4688 87.3199 20.00 100.00

1.00

2.00

3.00

4.00

5.00

Total

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval forMean

Minimum Maximum

The groups represent the five quintiles of school enrollment size, 1-281, 282-443, 444-570, 571-717, and 718-2968

SPSS EXAMPLE

ANOVA

g3rall00 Grade 3 Reading TAKS score 2000

717.444 4 179.361 1.317 .261

285.662 1 285.662 2.097 .148

365.725 1 365.725 2.685 .101

351.719 3 117.240 .861 .461

324.124 1 324.124 2.379 .123

314.910 1 314.910 2.312 .129

36.809 2 18.405 .135 .874

12.018 1 12.018 .088 .766

11.656 1 11.656 .086 .770

25.153 1 25.153 .185 .667

393424.8 2888 136.227

394142.2 2892

(Combined)

Unweighted

Weighted

Deviation

Linear Term

Unweighted

Weighted

Deviation

QuadraticTerm

Unweighted

Weighted

Deviation

Cubic Term

BetweenGroups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

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