HW #10: Path Analysis

2. Path Analysis of Theoretical Model A

Before diving into the analysis, I will load my packages, import the data, and take a look.

Packages

# Projects
library(here)

# R Markdown
library(knitr)
library(rbbt)

# Statistics
library(psych)

# Tidyverse
library(readxl)
library(tidyverse)
library(broom)
library(glue)
library(corrr)

Data

tbl.2 <- read_excel(here('data', 'AMBSTD.xlsx'))

tbl.2[1:10, ] %>%
  kable

amb	cd	tp	hi	anx
-1.8254544	-2.3190302	-2.2932604	-1.5294401	-1.9294029
1.8031440	1.6117188	0.6198821	0.2343117	1.0286644
-0.2481765	0.7392864	1.0568971	0.3556038	1.3984228
-0.2481765	-0.1335827	0.3285387	0.9948955	-0.8201277
1.1830218	1.1755026	0.7658451	0.3565158	-0.0806108
0.5628995	-0.5711088	1.0568971	1.4198739	0.6589060
0.0845604	-0.5711088	-1.2741412	-2.0155206	-0.4503692
0.2287305	-0.1348926	-0.3995284	-1.0424476	-0.4503692
1.5167134	0.7388497	0.7658451	0.2042167	3.9867317
1.6599287	0.3030702	-0.1084764	-0.8290464	-0.8201277

describe(tbl.2)[c('n', 'mean', 'sd', 'min', 'max')] %>%
  kable

	n	sd	min	max
amb	211	1	-3.497254	1.803144
cd	211	1	-3.628115	1.611719
tp	211	1	-2.293843	1.493912
hi	211	1	-2.259017	2.149451
anx	211	1	-1.929403	3.986732

No data are missing, and the data are fully standardized. Excellent.

a. Path Coefficients

First, I’m going to write down Model A as a system of equations, using equations 15.1a-15.1d in Pedhazur (1982) .

\[ \begin{aligned} AMB &= e_1 \\ CD &= p_{21}AMB + e_2 \\ TP &= p_{31}AMB + p_{32}CD + e_3 \\ HI &= p_{42}CD + p_{43}TP + e_4 \\ ANX &= p_{53}TP + p_{54}HI + e_5 \end{aligned} \] These equations will inform the regression models I estimate.

I believe, because AMB is an exogenous variable, the estimate of \(e_1\) is simply the expected value of AMB.

e1 <- mean(tbl.2$amb)
e1

## [1] 1.266381e-15

Now for the interesting path coefficients.

fit.2.2 <- lm(cd ~ amb, tbl.2)
fit.2.2 %>%
  tidy %>%
  kable

term	estimate	std.error	statistic	p.value
(Intercept)	0.0000000	0.0575689	0.00000	1
amb	0.5513968	0.0577058	9.55531	0

\(p_{21}\) is the estimate of the coefficient of amb, so \(p_{21} \approx 0.55\).

According to Pedhazur (1982), \(e_j = \sqrt{1 - R^2_{j,12...i}}\), “where \(R^2_{j,12...i}\) is the squared multiple of correlation of endogenous variable \(j\) with variables \(1, 2, \dots, i\)” (Pedhazur, 1982, p. 585). Thus, in the single-predictor, standardized case, \[ \begin{aligned} e_2 &= \sqrt{1-R^2_{2,1}} \\ &= \sqrt{1-r^2_{21}} \\ &= \sqrt{1-p^2_{21}} \end{aligned} \]

p21 <- coef(fit.2.2)["amb"]

e2 <- sqrt(1 - p21^2)
e2

##       amb 
## 0.8342431

The result is that \(e_2 \approx 0.83\).

Next, let’s take a look at \(p_{31}\), \(p_{32}\) and \(e_3\).

fit.2.3 <- lm(tp ~ amb + cd, tbl.2)
p31 <- coef(fit.2.3)["amb"]
p32 <- coef(fit.2.3)["cd"]
fit.2.3 %>%
  tidy %>%
  kable

term	estimate	std.error	statistic	p.value
(Intercept)	0.0000000	0.0647161	0.000000	1.0000000
amb	0.2171172	0.0777591	2.792176	0.0057235
cd	0.1834380	0.0777591	2.359054	0.0192481

\(p_{31}\) is the effect of AMB on TP, and is approximately .22. \(p_{32}\) is the effect of CD on TP, and is approximately .18.

\(e_3\) is slightly more complicated to calculate than \(e_2\) because we now have more than one predictor. Thankfully, R can automatically compute \(R^2\) for the fitted model.

R23 <- summary(fit.2.3)$r.squared
e3 <- sqrt(1 - R23)
e3

## [1] 0.9355688

Thus, \(e_3 \approx .94\), which is a little larger than \(e_2\).

The process for \(p_{42}\), \(p_{43}\), \(e_4\), \(p_{53}\), \(p_{54}\), and \(e_5\) is essentially the same as that presented above, so I will present the syntax and results with less commentary.

fit.2.4 <- lm(hi ~ cd + tp, tbl.2)
p42 <- coef(fit.2.4)["cd"]
p43 <- coef(fit.2.4)["tp"]
e4 <- sqrt(1 - summary(fit.2.4)$r.squared)

fit.2.5 <- lm(anx ~ tp + hi, tbl.2)
p53 <- coef(fit.2.5)["tp"]
p54 <- coef(fit.2.5)["hi"]
e5 <- sqrt(1 - summary(fit.2.5)$r.squared)

tibble(p42, p43, e4, p53, p54, e5) %>%
  kable

p42	p43	e4	p53	p54	e5
0.1859613	0.3846249	0.8798383	0.2104845	0.3135576	0.8939613

Now with all the coefficients estimated, I can point out that \(p_{21}\) (the effect of AMB on CD) is the strongest, followed by \(p_{43}\).

b. Decomposed Correlations

Definitions of direct effect, indirect effect, total effect, spurious component, and unanalyzed component Pedhazur (1982):

Direct Effect: The estimated effect of one variable on another.
- Ex: The effect of AMB on CD.
Indirect Effect: The mediated effect of one variable on another.
- Ex: The effect of AMB on HI via CD.
Total Effect (also known as the effect coefficient): The sum of the direct and indirect effects of one variable on anther.
- Ex: The effect of AMB on TP and the effect of AMB on TP via CD.
Spurious Component: The correlation between two variables not explained by the total effect of one variable on another&em;in other words, the correlation between two variables that is explained by a common cause.
- Ex: The correlation between CD and TP explained by AMB, rather than the effect of CD on TP.
Unanalyzed Component: The correlation between two variables not explained by a causal path in the model.
- Ex: The effect of AMB on TP explained by AMB’s correlation with CD and CD’s effect on TP in Theoretical Model B. No correlations in Theoretical Model A will have unanalyzed components.

The sum of the spurious and unanalyzed components is the noncausal part of the correlation coefficient.

Let’s move on to the calculations, beginning with \(r_{12}\), the correlation between AMB and CD.

\(r_{12}\) is fully explained by \(p_{21}\), the direct effect of AMB on CD, which we already know is approximately .55.

DE.12 <- p21
r12 <- DE.12

\(r_{13}\), the correlation between AMB and TP, is a combination of the direct effect of AMB on TP and the indirect effect mediated by CD. Thus, \(r_{13} = p_{31} + p_{21}p_{32}\).

DE.13 <- p31
IE.13 <- p21 * p32
r13 <- DE.13 + IE.13

\(r_{23}\), the correlation between CD and TP has a direct effect and a spurious component: \(r_{23} = p_{32} + p_{21}p_{31}\).

DE.23 <- p32
S.23 <- p21 * p31
r23 <- DE.23 + S.23

Moving along, \(r_{14}\), the correlation between AMB and HI, is the sum of three indirect effects, mediated by CD, TP, and the combination of both CD and TP: \(r_{14} = p_{21}p_{42} + p_{31}p_{43} + p_{21}p_{32}p_{43}\).

IE.14 <- p21 * p42 + p31 * p43 + p21 * p32 * p43
r14 <- IE.14

\(r_{24}\) is composed of a direct effect, an indirect effect, and a spurious component.

DE.24 <- p42
IE.24 <- p32 * p43
S.24 <- p21 * p31 * p43
r24 <- DE.24 + IE.24 + S.24

\(r_{34}\) is composed of a direct effect and a spurious component. Because the spurious component is solely a result of TP’s correlation with CD, I will reuse \(r_{23}\).

DE.34 <- p43
S.34 <- r23 * p42
r34 <- DE.34 + S.34

\(r_{15}\) comprises only an indirect effect. Well, really five indirect effects.

IE.15 <- p31 * p53 + p21 * p42 * p53 + p31 * p43 * p53 + p21 * p32 * p53 + p21 * p32 * p43 * p54
r15 <- IE.15

\(r_{25}\) comprises both an indirect effect and a spurious component.

IE.25 <- p32 * p53 + p42 * p54 + p32 * p43 * p54
S.25 <- p21 * p31 * p53 + p21 * p31 * p43 * p54
r25 <- IE.25 + S.25

\(r_{35}\) comprises a direct effect, an indirect effect, and a spurious component.

DE.35 <- p53
IE.35 <- p43 * p54
S.35 <- r23 * p42 * p54
r35 <- DE.35 + IE.35 + S.35

\(r_{45}\) is the final correlation to decompose. It comprises a direct effect and a spurious component.

DE.45 <- p54
S.45 <- r34 * p53
r45 <- DE.45 + S.45

All the components are broken down in the following table.

tbl.2.b <- tibble(Vars = c("AMB, CD",
                "AMB, TP", "CD, TP",
                "AMB, HI", "CD, HI", "TP, HI",
                "AMB, ANX", "CD, ANX", "TP, ANX", "HI, ANX"),
       r = c(r12, r13, r23, r14, r24, r34, r15, r25, r35, r45),
       DE = c(DE.12, DE.13, DE.23, NA, DE.24, DE.34, NA, NA, DE.35, DE.45),
       IE = c(NA, IE.13, NA, IE.14, IE.24, NA, IE.15, IE.25, IE.35, NA),
       S = c(NA, NA, S.23, NA, S.24, S.34, NA, S.25, S.35, S.45),
       U = c(rep(NA,10)))

tbl.2.b %>% kable

Vars	r	DE	IE	S	U
AMB, CD	0.5513968	0.5513968	NA	NA	NA
AMB, TP	0.3182644	0.2171172	0.1011471	NA	NA
CD, TP	0.3031557	0.1834380	NA	0.1197177	NA
AMB, HI	0.2249509	NA	0.2249509	NA	NA
CD, HI	0.3025626	0.1859613	0.0705548	0.0460464	NA
TP, HI	0.4410001	0.3846249	NA	0.0563752	NA
AMB, ANX	0.1183483	NA	0.1183483	NA	NA
CD, ANX	0.1586804	NA	0.1190434	0.0396369	NA
TP, ANX	0.3487634	0.2104845	0.1206021	0.0176769	NA
HI, ANX	0.4063813	0.3135576	NA	0.0928237	NA

c. Fit Assessment

Overall Fit

In Theoretical Model A, we have have an “overidentified model” (Pedhazur, 1982, p. 617) because there are fewer paths in the model than are possible. This is equivalent to saying that the model implicitly hypothesizes no effect of some variables on others.

Specifically, the table above clearly shows that AMB has no direct effect on HI or ANX, and CD has no direct effect on ANX. Thus, in this case, we have three “overidentifying restrictions” (Pedhazur, 1982, p. 618), which will equal the degrees of freedom in the significance test.

\[d = df = 3\] In order to calculate the test statistic, \(W\), in addition to \(d\), we need the sample size, \(N\), the generalized squared multiple correlation, \(R^2_m\), and what I call ‘Specht’s M’. Specht’s M is essentially a rebranded \(R^2_m\) for overidentified models Specht (1975).

The exact formula is Equation 15.21 in (1982, p. 619). I present it here with modernized notation:

\[W = - (N - d)\ln{\frac{1 - R^2_m}{1-M}}\] \(W\) is approximately \(\chi^2\)-distributed with \(df = d\) degrees of freedom.

I will now calculate each variable in turn.

N <- nrow(tbl.2)
d <- 3

If I understand this correctly, \(R^2_m\) is simply the product of (one minus) all the squared observed correlations, less than one.

# Note: Using for loops in R is discouraged, but it's not a big deal for a short vector, 
# and using nested map() functions is annoying.
num.2.c <- c()
for (col in select(correlate(tbl.2), -c(1, 2))) {
  for (r in col) {
    if (!is.na(r)) {
      num.2.c <- c(num.2.c, r)
    } else {
      break
    }
  }
}

## 
## Correlation method: 'pearson'
## Missing treated using: 'pairwise.complete.obs'

R2m <- 1 - prod(map_dbl(num.2.c, ~ 1 - . ^ 2))

Finally, I believe Specht’s \(M\) is calculated in the same way, except by using the reproduced correlations instead of the observed correlations.

M <- 1 - prod(map_dbl(tbl.2.b$r, ~ 1 - . ^ 2))

Now, I can calculate \(W\) and test it against a \(\chi^2\) distribution.

W <- -(N - d) * log((1 - R2m) / (1 - M))
p <- dchisq(W, d)
tibble(W = W, p = p) %>%
  kable

W	p
2.152695	0.199503

This result suggests that the model is not a good fit.

Correspondence Between Observed and Reproduced Correlations

To compare the observed correlations and the reproduced correlations, we can simply start by taking the differences.

tbl.2.c <- tbl.2.b %>%
  select(c(1,2)) %>%
  rename(Reproduced = r) %>%
  mutate(Observed = num.2.c, .after = Vars) %>%
  mutate(Difference = Observed - Reproduced)

tbl.2.c %>%
  kable

Vars	Observed	Reproduced	Difference
AMB, CD	0.5513968	0.5513968	0.0000000
AMB, TP	0.3182644	0.3182644	0.0000000
CD, TP	0.3031557	0.3031557	0.0000000
AMB, HI	0.2813076	0.2249509	0.0563567
CD, HI	0.3025626	0.3025626	0.0000000
TP, HI	0.4410001	0.4410001	0.0000000
AMB, ANX	0.1122043	0.1183483	-0.0061440
CD, ANX	0.0821677	0.1586804	-0.0765127
TP, ANX	0.3487634	0.3487634	0.0000000
HI, ANX	0.4063813	0.4063813	0.0000000

It appears that for all the pairs of variables with direct effects, the difference between the observed and reproduced correlations is infinitesimally small. I’m not sure what to make of that. It may simply be a result of those relationships being identified. The differences for the variables without direct effects are larger, but without a significance test, it’s difficult to say how much credence to lend the differences.

Although we are not talking about correlations from two samples, it might be worth doing a Fisher \(r\)-to-\(z\) transformation and checking whether those three correlations are significantly different.

tbl.2.c %>%
  select(- "Difference") %>%
  mutate(z = paired.r(Observed, Reproduced, n = nrow(tbl.2))$z,
         p = paired.r(Observed, Reproduced, n = nrow(tbl.2))$p) %>%
  kable

Vars	Observed	Reproduced	z	p
AMB, CD	0.5513968	0.5513968	0.0000000	1.0000000
AMB, TP	0.3182644	0.3182644	0.0000000	1.0000000
CD, TP	0.3031557	0.3031557	0.0000000	1.0000000
AMB, HI	0.2813076	0.2249509	0.6142955	0.5390201
CD, HI	0.3025626	0.3025626	0.0000000	1.0000000
TP, HI	0.4410001	0.4410001	0.0000000	1.0000000
AMB, ANX	0.1122043	0.1183483	0.0635007	0.9493678
CD, ANX	0.0821677	0.1586804	0.7921770	0.4282575
TP, ANX	0.3487634	0.3487634	0.0000000	1.0000000
HI, ANX	0.4063813	0.4063813	0.0000000	1.0000000

If I did this right, then none of the three are even remotely significantly different.

d. Hypotheses

Tested as a whole, Theoretical Model A did not perform well. It did not explain enough additional variance vs. an identified model to justify the additional constraints.

The tests of individual correlations tell the same story. Constraining the three pairs of variable to indirect paths did not lead to improved reproduced correlations vs. the observed correlations.

Thus, we cannot conclude that the omitted paths should be rejected from the model.

e. Reliabilities

I’m sure we will find out exactly what implications reliabilities have when we move onto SEM and include a measurement model. For now, suffice it to say that these relatively low scores imply that an assumption of path analysis may have been violated (Pedhazur, 1982, p. 633).

My guess is that in this case, the low reliabilities are more likely to bias the effect sizes downward, but without more information about the measures, it’s difficult to say.

f. Conclusions

I’m going to go out on a limb and say “ambition” is related to “competitive drive” and “time pressure” is related to “hurried/impatient”, although I hardly think we needed this study to tell us that. Each is, if not synonymous, then at least closely semantically related to the other. But the data do bear those relationships out.

Overall, I don’t think this analysis lends Theoretical Model A much support.

3. Path Analysis of Theoretical Model B

Theoretical Model B resolves one of my complaints, namely that ambition did not seem so much like a cause of competitive drive as a conceptually-related construct. Let’s see if that makes a difference.

a. Path Coefficients

With the exception of \(p_{21}\)’s theoretical replacement with \(r_{12}\) (which has the identical value), all the path coefficients remain the same to those in Part 2. Therefore, please refer to Part 2.a for the path coefficients.

However, as we shall see in a moment, the decomposed correlations will be different because some previously analyzed components will no longer be so.

b. Decomposed Correlations

I will go in the same order as before.

\(r_{12}\) is simply the unanalyzed “path” between AMB and CD, represented by a correlation.

# this seems almost absurdly unnecessary, but I'm including it for the sake
# of completeness
U.12 <- r12
r12 <- U.12

\(r_{13}\) includes the direct effect of AMB on TP and an unanalyzed component for the effect of CD on TP.

DE.13 <- p31
U.13 <- p32 * r12
r13 <- DE.13 + U.13

Likewise, \(r_{23}\) comprises a direct effect of CD on TP and an unanalyzed component for the effect of AMB on TP.

DE.23 <- p32
U.23 <- p31 * r12
r23 <- DE.23 + U.23

Moving along, \(r_{14}\) comprises an indirect effect and an unanalyzed component.

IE.14 <- p31 * p43
U.14 <- (p42 + p32 * p43) * r12
r14 <- IE.14 + U.14

So many unanalyzed components! \(r_{24}\) comprises a direct effect, an indirect effect, and an unanalyzed component.

DE.24 <- p42
IE.24 <- p32 * p43
U.24 <- p31 * p43 * r12
r24 <- DE.24 + IE.24 + U.24

Next, \(r_{34}\) has a direct effect, a spurious component (thanks to the common cause of CD), and an unanalyzed component.

DE.34 <- p43
S.34 <- p32 * p42
U.34 <- p31 * p42 * r12
r34 <- DE.34 + S.34 + U.34

Getting close… \(r_{15}\) is composed of an indirect effect and an unanalyzed component.

IE.15 <- p31 * p53 + p31 * p43 * p54
U.15 <- (p32 * p53 + p42 * p53 + p32 * p43 * p54) * r12
r15 <- IE.15 + U.15

\(r_{25}\) is similarly composed of an indirect effect and an unanalyzed component. The \(r_{12}\) is simply switched.

IE.25 <- p32 * p53 + p42 * p54 + p32 * p43 * p54
U.25 <- (p31 * p53 + p31 * p43 * p54) * r12
r25 <- IE.25 + U.25

Almost there… \(r_{35}\) comprises a direct effect, an indirect effect, a spurious component, and an unanalyzed component. Wow!!!!

DE.35 <- p53
IE.35 <- p43 * p54
S.35 <- p32 * p42 * p54
U.35 <- p31 * p42 * p54 * r12
r35 <- DE.35 + IE.35 + S.35 + U.35

Last and arguably least, \(r_{45}\) comprises a direct effect, a spurious component, and an unanlyzed component.

DE.45 <- p54
S.45 <- p42 * p32 * p53 + p43 * p53
U.45 <- (p42 + p32 * p43) * (p31 * p53) * r12
r45 <- DE.45 + S.45 + U.45

As before, I’ve broken down all the components in a single table.

tbl.3.b <- tibble(Vars = c("AMB, CD", "AMB, TP", "CD, TP",
                           "AMB, HI", "CD, HI", "TP, HI",
                           "AMB, ANX", "CD, ANX", "TP, ANX", "HI, ANX"),
       r = c(r12, r13, r23, r14, r24, r34, r15, r25, r35, r45),
       DE = c(NA, DE.13, DE.23, NA, DE.24, DE.34, NA, NA, DE.35, DE.45),
       IE = c(NA, IE.13, NA, IE.14, IE.24, NA, IE.15, IE.25, IE.35, NA),
       S = c(NA, NA, NA, NA, NA, S.34, NA, NA, S.35, S.45),
       U = c(U.12, U.13, U.23, U.14, U.24, U.34, U.15, U.25, U.35, U.45))

tbl.3.b %>% kable

Vars	r	DE	IE	S	U
AMB, CD	0.5513968	NA	NA	NA	0.5513968
AMB, TP	0.3182644	0.2171172	0.1011471	NA	0.1011471
CD, TP	0.3031557	0.1834380	NA	NA	0.1197177
AMB, HI	0.2249509	NA	0.0835087	NA	0.1414422
CD, HI	0.3025626	0.1859613	0.0705548	NA	0.0460464
TP, HI	0.4410001	0.3846249	NA	0.0341124	0.0222629
AMB, ANX	0.1269558	NA	0.0718846	NA	0.0550712
CD, ANX	0.1586804	NA	0.1190434	NA	0.0396369
TP, ANX	0.3487634	0.2104845	0.1206021	0.0106962	0.0069807
HI, ANX	0.4081592	0.3135576	NA	0.0881377	0.0064639

c. Fit Assessment

Overall Fit

I will attempt to use the same procedures as before. But first, it is necessary to determine whether the proposed correlation (and not causation) between AMB and CD constitutes an overidentifying restriction. (Pedhazur, 1982), to my reading, is ambiguous on this point. Instead, I relied on (Kaplan, 2009)’s explication of Kenneth Bollen’s counting rule. This clarifies that the correlation between exogenous variables is not parameterized. Thus, the correlation is an overidentifying restriction, and \(d\) increases from 3 to 4.

d <- 4

\(N\) remains the same at 211. However, I believe the values of \(R^2_m\) and Specht’s \(M\) will have changed.

Repurposing my code from earlier (another R “no-no”):

num.3.c <- c()
for (col in select(correlate(tbl.2), -c(1, 2))) {
  for (r in col) {
    if (!is.na(r)) {
      num.3.c <- c(num.3.c, r)
    } else {
      break
    }
  }
}

## 
## Correlation method: 'pearson'
## Missing treated using: 'pairwise.complete.obs'

R2m <- 1 - prod(map_dbl(num.3.c, ~ 1 - . ^ 2))

M <- 1 - prod(map_dbl(tbl.3.b$r, ~ 1 - . ^ 2))

W <- -(N - d) * log((1 - R2m) / (1 - M))

p <- dchisq(W, d)

tibble(W = W, p = p) %>%
  kable

W	p
1.33921	0.171389

The result is that \(W\) is lower than before, but it is still not statistically significant.

Correspondence Between Observed and Reproduced Correlations

At this point, I will check whether the reproduced correlations for the three restricted paths (not including the correlation between AMB and CD) are significantly different from the observed correlations.

tbl.2.b %>%
  select(c(1,2)) %>%
  rename(Reproduced = r) %>%
  mutate(Observed = num.2.c, .after = Vars) %>%
  mutate(z = paired.r(Observed, Reproduced, n = nrow(tbl.2))$z,
         p = paired.r(Observed, Reproduced, n = nrow(tbl.2))$p) %>%
  kable

Vars	Observed	Reproduced	z	p
AMB, CD	0.5513968	0.5513968	0.0000000	1.0000000
AMB, TP	0.3182644	0.3182644	0.0000000	1.0000000
CD, TP	0.3031557	0.3031557	0.0000000	1.0000000
AMB, HI	0.2813076	0.2249509	0.6142955	0.5390201
CD, HI	0.3025626	0.3025626	0.0000000	1.0000000
TP, HI	0.4410001	0.4410001	0.0000000	1.0000000
AMB, ANX	0.1122043	0.1183483	0.0635007	0.9493678
CD, ANX	0.0821677	0.1586804	0.7921770	0.4282575
TP, ANX	0.3487634	0.3487634	0.0000000	1.0000000
HI, ANX	0.4063813	0.4063813	0.0000000	1.0000000

Once again, the reproduced correlations are not significantly different from the observed correlations.

d. Hypotheses

Theoretical Model B may have performed better than Theoretical Model A, but not sufficiently to be able to say that it performed well. The overall fit assessment found it be a non-significant improvement on the identified model. And the reproduced correlations in Theoretical Model B were identical to those in Theoretical Model A (this is worrying to me, but I can’t see how I might have made a mistake in calculating the reproduced correlations).

e. Conclusions

Because Theoretical Model B did perform (if nonsignficantly, at least directionally) better than Theoretical Model A, I will take it at face value that the respecification of the relationship between AMB and CD is justified. In other words, it’s better to say that ambition and competitive drive are correlated than that ambition causes competitive drive.

However, I can’t help but wonder if a Theoretical Model C in which the relationship between time pressure and hurried/impatient were respecified might perform even better. The two seem closely related, and although we generally conceive of cognitive states as leading to behaviors, I imagine that the measures do not precisely distinguish between the two.

References

bbt_write_bib("biblio.json", c("pedhazur82", "specht75", "kaplan09"), overwrite = TRUE)

Kaplan, D. (2009). Structural Equation Modeling (2nd ed.): Foundations and Extensions. SAGE Publications, Inc. https://doi.org/10.4135/9781452226576

Pedhazur, E. J. (1982). Path Analysis. In Multiple Regression in Behavioral Research. Holt.

Specht, D. A. (1975). On the evaluation of causal models. Social Science Research, 4(2), 113–133. https://doi.org/10.1016/0049-089X(75)90007-1

HW #10: Path Analysis

Daniel Lewis

7/28/2020

2. Path Analysis of Theoretical Model A

Packages

Data

a. Path Coefficients

b. Decomposed Correlations

c. Fit Assessment

Overall Fit

Correspondence Between Observed and Reproduced Correlations

d. Hypotheses

e. Reliabilities

f. Conclusions

3. Path Analysis of Theoretical Model B

a. Path Coefficients

b. Decomposed Correlations

c. Fit Assessment

Overall Fit

Correspondence Between Observed and Reproduced Correlations

d. Hypotheses

e. Conclusions

References