Stat404
Fall 2009
Lab 11
1. Consider the following path model based on a sample of
250 subjects (Ss):
a. Although a “just identified” path model need not be “fully
recursive,” all “fully recursive” path models are “just identified.”
What do these two expressions mean? (In defining each expression, be
sure to explain how you can tell that these expressions apply to
this path model.)
b. Whenever a path model is fully recursive, you are able to
reproduce the original correlation matrix from the paths. Below is
a correlation matrix among Z1 (Gender), Z2 (Race), Z3 (S’s
education), Z4 (S’s occupation), and Z5 (S’s present income).
Missing in this matrix are all correlations with Z5. Using the
values of paths in the above path model, calculate these four
missing correlations (identified in the matrix with question marks).
Show your work.
Z1
Z2
Z3
Z4
Z1
1.0
Z2
.30
1.0
Z3
.25
.15
1.0
Z4
.30
.35
.40
1.0
Z5
?
?
?
?
Z5
1.0
c. The model does not include coefficients for the residual paths to
S's education, S's occupation, or S's income. Calculate these.
1
d. Calculate the total, direct, and indirect effects of
the exogenous variables and endogenous variables on the
endogenous variables. Present these effects in a table that
shows that the total effects equal the sums of the direct
and indirect effects.
e. Delete the paths from race to S's education and from
gender to S's present income and draw the revised path
diagram. (In Lab 12 this restricted model will be compared to the
fully recursive model for goodness of fit.)
Some help for part e: The following program illustrates how to regress
Y on W and X when you only know that rYW = .3 , rYX = .25 , and rWX
= .2 in a sample of n = 100 . Please note that you will need to
modify this program to obtain some of the paths called for in part e:
matrix data vars=y,w,x / contents=mean sd corr / n=100.
begin data.
0
0
0
1
1
1
1.00
.30 1.00
.25 .20 1.00
end data.
regression matrix=in(*) / var=y,w,x / dep=y / enter.
2. There are two conflicting theories on the development of
Post-Traumatic Stress Syndrome (a psychological disorder in which
American soldiers who fought in the Iraq War have later difficulties in
adjusting to civilian life). According to one theory (Theory A), a
soldier's strict family background (X1) prevents the soldier from
having a relativistic world view (X2). The theory continues to argue
that without a relativistic world view, soldiers who fought in Iraq
develop symptoms of the syndrome (X3). A less parsimonious theory
(Theory B) is entirely in agreement with Theory A, but also notes that
people who grow up in strict families often have symptoms of the
syndrome. (For example, people from strict family backgrounds are
known to have sudden uncontrollable outbursts of anger).
You collect data on 120 veterans of the War in Iraq and (using factor
analysis) you develop the following three measures:
X1 = Strictness of family background (high scores are strict).
X2 = Relativity of world view (high scores are relativistic).
X3 = Degree to which Post-Traumatic Stress Syndrome is manifest (high
scores reflect the greatest manifestation of the syndrome)
2
Correlations among these variables are . . .
X1
X2
X3
X1
1.0
-.30
.10
X2
-.30
1.0
.40
X3
.10
.40
1.0
a. Draw a path model for Theory A that includes all path and
residual coefficients. (Show your calculations!)
b. Draw a path model for Theory B that includes all path and
residual coefficients. (Show your calculations!)
c. Construct a table of effects for EACH path model.
3. An industrial psychologist is investigating the determinants of job
performance. She has data on four variables from a sample of 100
married employees of the American Widget Company. The variable names
are . . .
PERFORM:
COMMIT:
HAPPY:
KIDS:
High scores indicate high job performance
High scores indicate high job commitment
High scores indicate high marital happiness
Number of children under 18 years of age in the employee's
family
Correlations among these variables are . . .
PERFORM
COMMIT
HAPPY
KIDS
PERFORM
1.00
.65
.20
.30
COMMIT
.65
1.00
-.25
-.20
HAPPY
.20
-.25
1.00
-.03
KIDS
.30
-.20
-.03
1.00
a. It is well-established in previous research that job commitment
enhances (i.e., increases) job performance. It is also
well-established that job commitment is less common among employees
with happy marriages than with unhappy marriages. (This is due to
happily married employees' displacement of commitment from their
jobs to their marriages.) Using data presented in the correlation
matrix, draw a path model that depicts these well-established
relations among PERFORM, COMMIT, and HAPPY. (Be sure to include the
numerical values for all relevant paths and for all residual
coefficients as well as to indicate how you arrived at these
values.)
3
b. What has not been established in previous research is
whether the number of an employee's children has an
effect on the well-established process described in part a.
In particular, the industrial psychologist theorizes that,
although job performance is likely to be greater among
employees with increasingly larger numbers of children (due
to the increasing feeling of family responsibility that
parenthood brings), the total effect of employees' number
of children will be of a decrease in job performance, because
children (like spouses in happy marriages) entice displacement of
commitment from employees' jobs to their families. To test the role
of KIDS in the causal relations among HAPPY, COMMIT, and PERFORM,
the path model in part a was extended to a model that included
statistics from output generated by the following two SPSS commands:
regression vars=COMMIT,HAPPY,KIDS / dep=COMMIT / enter
regression vars=PERFORM,COMMIT,KIDS / dep=PERFORM / enter
Part of the output from the first regression is . . .
------------------ VARIABLES IN THE EQUATION -----------------VARIABLE
B
SE B
BETA
KIDS
HAPPY
(constant)
-4.153738
-.512461
89.055150
1.921351
.192135
12.408583
-.207687
-.256231
Part of the output from the second regression is . . .
------------------ VARIABLES IN THE EQUATION -----------------VARIABLE
B
SE B
BETA
KIDS
COMMIT
(constant)
13.437500
1.109375
-42.343750
1.928730
.096436
7.007529
.447917
.739583
In her extended path model, the researcher decided not to specify a
causal direction in the relation between KIDS and HAPPY. Draw the
path model that depicts this unanalyzed relation, the causal
relations that the researcher estimated in the above regressions,
and the well-established causal relations depicted in the path model
drawn in part a. (Again, be sure to include the numerical values for
all relevant paths and for all residual coefficients as well as to
indicate how you arrived at these values.)
4
c. Based on the path model drawn in part b, what is the
direct effect of KIDS on PERFORM? (Be sure to indicate how
you arrived at your answer.)
d. Based on the path model drawn in part b, what is the
indirect effect of KIDS on PERFORM? (Be sure to indicate how
you arrived at your answer.)
Below please find R and SAS code for problem 1.e.:
# R
# Code:
nvars <- 3
#----Enter the number of variables----#
corrs <- matrix(c
(1.00, 0.30, 0.25, #----Enter the correlation matrix with----#
0.30, 1.00, 0.20, #----Y in column 1 and row 1.----#
0.25, 0.20, 1.00)
, ncol=nvars, byrow=T)
R <- corrs[2:nvars,2:nvars]
#----Create matrix R of corrs excluding
column 1 and row 1----#
r <- corrs[1,2:nvars] #----Create vector r of corrs in first row
excluding the first value----#
beta.hats <- solve(R) %*% r #----The inverse of R multiplied by r----#
beta.hats
* SAS
* Code;
PROC IML;
corrs = {1.00 0.30 0.25, /*****Enter correlation matrix with Y in
0.30 1.00 0.20,
column 1 and row 1*****/
0.25 0.20 1.00};
Rbig = corrs[{2 3}, {2 3}]; /*****Create submatrix R consisting of
rows 2-3 and columns 2-3*****/
rsmall = corrs[{1}, {2 3}]; /*****Create submatrix r consisting of
row 1, columns 2-3*****/
Rbig_inv = inv(Rbig);
/***** Inverse of Rbig *****/
rsmall_trans = (rsmall)`;
/***** Inverse of rsmall *****/
betas = rbig_inv*rsmall_trans; /***** Matrix muliplication *****/
5
PRINT betas;
RUN;
6