09 SOC 681Estimating and Testing Hypotheses about Means

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

,var_cue

recall1

cued1

cov_rc

Example 13: Model A
Homogenous covariance structures
Attig (1983) young subjects
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 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 B

Invariant means and (co-)variances
Attig (1983) young subjects
Model 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

knowledge
0,

value

performance

1

error

satisfaction

Example 14
Job Performance of Farm Managers
Regression 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

verbal

visperc

1

cube_s

lozn_s

mn_v,

int_vis
1

1
sent_v

word_v

int_cub
1

cubes

int_loz

1

lozenges

int_par
1

paragrap

int_sen
1

sentence

int_wrd
1

wordmean

0,

err_v
0,

err_c
0,

err_l
0,

err_p
0,

err_s
0,

err_w

Example 15: Model A
Factor analysis with structured means

Holzinger and Swineford (1939): Girls' sample
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 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