Ancestry Estimation - Research Methods in Human Skeletal Biology - page 133

Ordinal Regression
Ordinal regression analysis (ORA) measures the association of an ordinal response vari-
able (a categorical variable with ordering d i.e., small, medium, large) to a set of predictor
variables (a variable used to predict the value of another variable). In traditional linear
regression, the sum-of-squared differences between a continuous dependent variable
and the weighted combination of the independent variables are minimized prior to calcu-
lating regression coefficients. This is not the case when the dependent variable is ordinal.
Ordinal regression calculates coefficients based on the assumption that the response vari-
able is a categorical response with some underlying continuous distribution. In most
cases, there is a valid theoretical basis for assuming this underlying distribution.
However, even when this assumption is not met, the model can still theoretically produce
valid results.
Rather than predicting the actual cumulative probabilities, an ORA predicts a function of
those values using a process known as a link function. Simplistically, the link function links
the model specified in the design matrix to the real parameters of the dataset. After initial
model development, the predicted probability of each response category can be used to
assign an unknown individual to a group. An ORA can be expressed as

link ðg ij Þ¼q j ½b 1 c i1 þ b 2 c i2 þ b p c ij (5.1)

where link( ) is the link function for the current analysis, g ij is the cumulative probability of
the jth category for the ith case, q j is the threshold for the jth category, p is the number of
regression coefficients,

c i1 . c ip are the values of the predictors for the ith case, and b 1 . b p
are the regression coefficients. One of the benefits of ORA, and a similarity of ORA to analysis
of variance (ANOVA), is the ability to assess the significance of individual response variables
and to test for any interaction between all response variables. For example, ORAs allow one
to determine if sex, ancestry, or the interaction of sex and ancestry significantly affect the
expression of inferior nasal aperture morphology.
Ordinal regression analysis can be carried out using the PLUM function in SPSS . The
purpose of the ORA in ancestry research is twofold. First, as mentioned above, the ORA
can be used to determine the significance of sex and ancestry, and the interaction of the
two, on the expression of each morphoscopic trait. Significance is assessed at the

0.05
level using the Wald statistic, a measure similar to the F-value in a traditional ANOVA.
Each of these parameter estimates is then assessed for significance. As an example, the
ORA parameter estimates for interorbital breadth are presented in Table 5.4 . Once all signif-
icant traits are determined, we can apply the ORA with all significant traits set as the

a ¼

TABLE 5.4 Parameter Estimates and Significance Levels for Interorbital Breadth

Ind. Variable Estimate Std. Error Wald df Sig.

Ancestry 2.492 0.340 53.723 1 0.000

Sex 1.113 1.250 0.792 1 0.373

Ancestry*Sex 0.929 1.299 0.512 1 0.420