Qualitative independent variables (also called covariates) can be binary (or dichotomous, 2 modalities only) or polytomous (3 or more modalities), and secondly they can be ordinal (order of modalities has a signification, ex : little/medium/big) or not.

Example (in Hosmer & Lemeshow) :
Low birth weight is harmful to chidren's growing, so a study was undertaken to evaluate riks of Low Birth Weignt (Y variable). Among 8 potential risk factors (independent variables), Authors of this study decided to keep the 4 following variables :

• 2 quantitative variables : age of the mother (AGE) and its weight et last menstrual period (LWT),
• 2 qualitative variables : number of first trimester physician visits (FTV) and race of the mother (RACE coded in 3 categories).

Parameters of the model are calculated through method of generalized least squares, each coefficient is analysed through test of Wald, and significance of the model is evaluated through method of maximum likelihood.

with :

• L₀ likelihood of the logistic regression model without covariates (i.e. only intercept)
• L_k likelihood of the logistic regression model with all covariates

This value follows the Khi² distribution.

1.5. Case of polytomous covariates in a logistic regression model

A polytomous covariate can show 3 (or more) different modalities. However, for calculations of coefficients of a logistic regression model, the modalities must be changed in numerical values. You can directely encode each modality ("0" for the first modality, "1" for the second modality, "2" for the third modality, etc...).

During the data selection procedure, StatEL allows you to specify the way each modality must be encoded. Nevertheles, in this situation, odds ratio measures the risk to have Y=1 when subjects show any modality compared to subjects who show the modality encoded in "0", without any other precision. Thus, in this way, information of modalities encoded in "1", "2", "3", ... is not fully exploited.

Nevertheles, there is another encoding method of a polytomous covariate that allows to calculate as many odds ratio as number of modalities (minus one : the reference modality), and thus to extract information of every modalities for this covariate. It consists in creation of new covariates: the "design variables", each one shows one modality of the initial polytomous covariate and is encoded in "0/1" (cf. below).

a) Coding of a polytomous variable in design variables:

Coding of a polytomous variable in design variables requires to choose which modality will be a "reference" compared to the others. Considering a qualitative variable with 4 modalities "A", "B", "C", "D". The last modality is the reference compared to the 3 others. Transformation of this polytomous variable leads to constitution of 3 new design variables, eash one is related to the 3 modalities, and encoded in "0/1" as follow :

Consequently, the logistic regression model contains 3 binary covariates instead of one polytomous covariate with 4 modalities. Each covariate has a coefficient whose exponential measures its odds ratio. Thus, according to example above, if the design variable N°1 has a coefficient b = 0.69, its odds ratio = 2 (= e^0.69). It means that the risk to have Y=1 is twice as often among subjects that show modality "A" than among subjects that show modality "D".

During the data selection procedure, StatEL manages creation of these design variables since you specify to do it. Besides coefficients and their significance, StatEL calculates their odds ratio as well as 95% confidence interval.

2.2. Selection of a numerical independent variable in the procedure of logistic regression with StatEL

After selection of the cells range containing data for the independent variable, StatEL checks if your selection contains exactely the same number of cells as for selection of the dependent variable. If not, an error message is displayed :

b) Encoding of a polytomous variable in design variables :

Choose the second option. The lower list activates in order to select the reference modality compared to the others. Since, the dialog box widens to inform you about the encoding way for design variables:

descriptive statistics for each independent variable,
the logistic regression model,
significance analysis of every coefficients of the model with test of Wald, as well as associated odds ratio and their 95% confidence interval,
significance analysis of the model with test of likelihood ratio,
tests of Pearson's Khi², deviance and Hosmer & Lemeshow,
plots of residuals and levers of different "covariate patterns".

In this example (in Hosmer & Lemeshow), the test of likelihood ratio shows that the calculated model contains at least one variable whose coefficient is significantely different from 0 (with a p-value < 0.0335). The test of Wald (table) on every coefficients show that only variables LWT (p < 0.0145) and RACE 2 (p < 0.0219) have an influence on value of the Y variable.

Their odds ratio are also significant (with a p-value < 0.05) since their 95% confidence intervals exclude the value 1. We conclude that the risk to give birth to low weight children is multiplicated by 0.9858 when the weight of the mother increases of 1 unit (by extension, we can calculate that this risk is multiplicated by e^{(10 x -0.01426)} = 0.867 when the weight of the mother increases of 10 units). Fruthermore, the risk to give birth to low weight children is multiplicated by 2.729 for mothers who belong to category RACE 2 compared to those belonging to reference category (RACE 1).

In this same example, the Pearson's Khi² test and the deviance test do not reject goodness of fit hypothesis. The test of Hosmer & Lemeshow suggests rejection goodness of fit hypothesis. Nevertheless, the associated comment recalls that powerful of this statistic is reduced when these conditions are not respected : N > 400 and estimated frequencies > 5 (cf. table of H&L). Since none of these conditions is respected, we shall not trus result of this statistic.