### Narrative interpretation of output results generated by various statistical softwares

Hypothetical example: We studied the relationship between occurrence of rain (no/yes) and environmental temperatures (continuous variable) (example 1) or type of season (summer or winter) (dichotomous variable) (example 2) in randomly selected 20 different cities (sample size) of world. Binary logistic regression analysis output in SPSS is shown below:

Above analysis involves Step 1 because we had a single independent variable with which it is not possible to further undertake stepwise forward or backward logistic regression. First cell also contains the name of independent predictor variable called temperature (in short Temp) given in our example 1. Let us further analyze and interpret all parameters given in the row of ‘Temp’. First parameter is named “B” and it is actually known as beta coefficient. What does beta coefficient mean and how do we interpret it practically? This is quite simple and explained below.

Beta coefficient, henceforth called B, is the probability of change in outcome variable for every one unit change in independent predictor variable. See, in above table value of B is 0.619 which indicates that for every one unit increase in environment temperature, the probability of occurrence of rain increases by 0.619 units. The point to be carefully noted here is about the sign of B- that is- whether it positive or negative integer. Had B been -0.619, it would have meant that for every one unit increase in environmental temperature, the probability of occurrence of rain would have been lowered by 0.619 units. Since B coefficient determines gradient of change (i.e. slope) in outcome variable against all possible values of predictor variable, it is also loosely named as ‘slope coefficient’. However, more technical name for B coefficient is “log odds ratio” and importantly, it differs from term “odds ratio” which we normally see in published literature. Indeed, “odds ratio” is given separately as “Exp (B)” in logistic regression output above only when asked for (not as a default command). Exp (B) means that we are going to display ‘exponent’ of ‘B coefficient’ or ‘log odds’. If you can recall your basic algebra of exponents and logarithm, you will remember that removal of logarithm from a given numeric value occurs by exponentiation function. Hence, usual ‘odds ratio’ is obtained in logistic regression by using exponent of ‘B coefficient or log odds ratio’.

Hypothetical example 2: Suppose, there were 99 elderly persons in a urological centre who were routinely screened for suspected carcinoma of prostate by measuring their prostatic serum antigen (PSA) levels. Subsequently all study participants underwent prostatic biopsy to confirm the diagnosis of prostatic cancer.

Stata Software output is shown below:

Above output initially describes the actual number of observations (sample size=99) and number of times study sample was re-sampled by bootstrapping (1000). Method of ROC estimation was non-parametric where status was diagnosis of prostatic cancer based on classifier called PSA (prostatic serum antigen values). AUC, area under curve, was 0.94 indicating high discriminatory accuracy with bootstrapped standard error and three types of corresponding 95% CI also provided. The point estimate (along with 95% CI) of AUC represented by ROC normally varies from 0.5 (diagonal line) to 1 (curve touching the left upper most corner). Rule of thumb is that greater the area under curve (AUC), stronger is the discriminatory ability of independent variable. An AUC of 0.5 means that ROC curve has no discriminatory ability. AUC becomes clinically meaningful if it crosses the value of 0.70. In our example, AUC for the outcome of prostatic cancer using a model with single predictor variable of prostatic serum antigen is 0.94. In simple terms, this AUC of 0.94 means that if there are 100 pairs of subjects, our independent variable of PSA will be able to correctly predict the subject with occurrence of cancer prostate in each of these 94 pairs.