Estimating and Testing Hypotheses about Means

James G. Anderson, Ph.D.

Purdue University

Estimating Means

• SEM is usually used to estimate variances, covariances and regression weights

• Example 13 demonstrates how to estimate and test hypotheses about means

• The data are from Attg-yng.xls and Attg_old.xls

,var_rec

recall1

,var_cue

cued1

cov_rc

Example 13: Model AHomogenous covariance structures

Attig (1983) young subjectsModel Specification

Analysis Properties Dialog Box

• Check the box for Estimate means and intercepts.

• The path diagram shows a means, variance pair of parameters for each exogenous variable.

• When you choose Calculate Estimates from the Analyze Menu, AMOS will estimate two means, two variances and a covariance for each group.

Results

• Chi Square = 4.588

• Df = 3

• Probability level = 0.205

Output

• Means (Young subjects/old subjects)

• Covariances (Young subjects/old subjects)

• Variances (Young subjects/old subjects)

mn_rec, var_rec

recall1

mn_cue, var_cue

cued1

cov_rc

Example 13: Model BInvariant means and (co-)variances

Attig (1983) young subjectsModel Specification

Results

• Chi Square = 19.267

• Df = 5

• Probability level = 0.002

Conclusions

• Hypothesis of equal variances and covariances is accepted

• Hypothesis of equal means is rejected

Regression with an Explicit Intercept

• SEM usually does not estimate the intercept for the linear equations

• Example 14 demonstrates how to estimate intercepts

• The data are from Warren5v.xls

value

knowledge

performance

satisfaction

0,

error1

Example 14Job Performance of Farm ManagersRegression with an explicit intercept

(Model Specification)

Analysis Properties Dialog Box

• Check the box for Estimate means and intercepts.

• The path diagram shows a means, variance pair of parameters for each exogenous variable.

• When you choose Calculate Estimates from the Analyze Menu, AMOS will estimate a mean for each predictor and the intercept for the linear equation.

Results

• Sample Moments:– 4 sample means– 4 sample variances– 6 sample covariances– Df= 14

• Parameters to be Estimated:– 3 means– 3 variances– 3 covariances– 3 regression weights– 1 intercept– 1 error variance– Total = 14

Factor Analysis with Structured Means

• SEM can not estimate the means of comm0on factors in a single-sample factor analysis

• Example 15 demonstrates how to estimate differences in factor means across populations

• The data are from Grnt_fem.sav and Grnt_mal.sav

mn_s,

spatial

int_vis

visperc

int_cub

cubes

int_loz

lozenges

int_wrd

wordmean

int_par

paragrap

int_sen

sentence

0,

err_v

0,

err_c

0,

err_l

0,

err_p

0,

err_s

0,

err_w

mn_v,

verbal

1

cube_s

lozn_s

1

sent_v

word_v

1

1

1

1

1

1

Example 15: Model AFactor analysis with structured means

Holzinger and Swineford (1939): Girls' sampleModel Specification

Analysis Properties Dialog Box

• Check the box for Estimate means and intercepts.

• The path diagram shows a means, variance pair of parameters for each exogenous variable.

• When you choose Calculate Estimates from the Analyze Menu, AMOS will estimate two means, two variances and a covariance for each group.

Procedure

• Constrain the intercepts to be equal across groups– Right click on one of the observed variables (e.g.,

visperc)– Choose Object Properties– Click the Parameters Tab– Enter a Parameter Name in the intercept text box– Select All Groups so that the intercept is named the

same in both groups– Continue in the same manner to give names to the

five other intercepts

Procedures

• Fix the factor means in one group at a constant. For example, fix the means of the boy’s spatial and verbal factors at 0.

• Next assign names to the girls’ factor means

Results

• Chi Square = 22.593

• Df = 24

• Probability level = 0. 544

Means for Girls

FACTOR Estimate SE CR Prob.

Spatial -1.066 0.881 -1.209 0.226

Verbal 0.956 0.521 1.836 0.066