12:1 Example of procedure in JMP.
IMPORTANT: this analysis has a simplified version of the exogenous variables that does not show the covariances due
to the various exogenous variables. The detailed procedure is in later chapters.
Three varieties of wheat were grown in Yugoslavia during 10 years. Each plot was scored for disease resistance
(DisRes) and lodging resistance (LodRes). Other variables measured were leaf area index (LAI), leaf area duration (LAD),
density of spikes per square meter (SPM), number of kernels per spike (KSP), and kernel weight (KWT).
Spike density, kernels per spike, and kernel weight are the components of yield. Because the diagram does not include
direct arrows from the treatments to yield, the effects of other factors, such as variety and year, on yield must take place
through effects on the three yield components or other unmeasured factors. First consider a simple path diagram where
the intermediate variables (LAI, LAD, DisRes and LodRes) are not included. Because variety and year are controlled
treatments, and all combinations are present, no correlation arrows are present among the direct effects and interactions
(the effects are orthogonal).
Path diagram showing three exogenous variables that reflect a factorial experiment of three wheat varieties by ten
SPM
Variety
KSP
Year
Yield
KWT
Variety
x
Year
Other
(resid.)
USPM
UKSP
UKWT
UOTH
years. Spikes per square meter (SPM), kernels per spike (KSP), and kernel weight (KWT) were measured as yield
components. Yield is analyzed as a function of the three measured components and the residual, which is attributed to
other components of yield and errors not captured by SPM, KSP or KWT. Each yield component and the residuals of yield
are analyzed by ANOVA as functions of the treatment effects, variety, year, and the interaction variety*year. In this
diagram, the total correlation between SPM and KSP is partitioned into 4 paths with the following steps between SPM and
KSP: Variety, year, variety*year, USPM UKSP. The numerical calculation of the parts is presented later.
MLR. First, Yield is regressed on SPM, KSP, and KWT. Save residuals to obtain ANOVA and correlations later. The
standardized partial regression coefficients for the predictors are the values of the arrow that join each one with Yield. The
path between U for yield and yield is the square root of 1-Rsquare=0.6873.
Continue by performing an ANOVA or multiple linear regression of each endogenous variable (variables that receive
arrows) as a function of all factors or variables that have arrows pointed at them. For example, SPM is analyzed by
ANOVA as a function of Variety, Year, and Variety*Year. The path information is completed by calculating all correlations
among variables through the multivariate platform. Include all continuous variables present in the path diagram. Do not
include the nominal variables Variety, Year and VarietyxYear in the multivariate analysis of correlations. The correlations
among endogenous variables can be calculated and added to the path, but these elements of the path are redundant
because the correlations are “contained” in all paths that join endogenous variables. For example, the correlation between
KSP and SPM is partitioned into components due to year, variety, interaction and correlations in the error terms. This will
be explained below.
ANOVA’s. Each yield component can be analyzed as a function of the year by variety factorial in one step with the Fit
Model platform as shown in the following figure. Make sure that the personality is Standard Least Squares, and that all
yield components are in the Y box.
We are interested in the proportion of the total SS explained by each effect in the model. The value of the path from
each effect to each yield component is the square root of the proportion of SS explained by the effect. For example, the
path from variety to SPM has a value of
pvar.SPM 
47850.2
 0.1481 . Calculate all paths, including those from
2180122.4
the error terms, following this example. Save the residuals for all three yield components.
pvar.SPM 
47850.2
2180122.4
Correlations. The correlation matrix and the level of significance of each pairwise correlation can be obtained through
the multivariate platform of JMP. To get the Pairwise correlation, select the option in the pop-down menu by clicking on the
red triangle on the left of “Multivariate” at the top of the results window. To complete the path we need all correlations
among yield, yield components, and residuals.
For easier
interpretation,
this table was
sorted by
probability level
by right-clicking
on it and
selecting “Sort.”
Note that the residuals of the yield components exhibit non-zero correlations with yield. These correlations arise from
the path from residuals to yield components and from yield components to yield, plus paths trough the correlations among
the residual for yield and the residuals for yield components. Note that the first correlation, between residual for yield and
yield, is 0.6873, a value that we had already calculated as 1 0.52761 . Also note that the correlations between the
residual for yield and each of yield’s explanatory variables (SPM, KSP and KWT) are zero.
Path diagram. Now we have all elements to complete the path diagram and then proceed to interpreting it. Values for
the double-headed arrows are obtained from the correlations table above. Direct paths are obtained form the ANOVA’s
and the MLR. The values inside boxes with borders are significant at the 0.01 probability level.
SPM
0.148
Variety
0.818
0.225
0.656
0.153
KSP
0.936
Year
0.713
0.455
Yield
0.661
0.429
0.870
KWT
0.164
0.254
Variety
x
Year
0.687
0.315
0.312
0.275
USPM
Other
(resid.
yield)
0.242
-0.487
0.182
0.349
UKSP
-0.048
UKWT
-0.114
-0.109
-0.701
0.181
Uyield