The coefficients are the maximum likelihood estimates of  β0 and β1. After fitting the model the next step is to interpret those coefficients. 

This time the question we need to answer is: "What do the estimates of the coefficients tell us about the research question?"

For the logistic regression model, from the logit which we will call g(x), we can write the following:

g(x) =  β0+ β1x1

where

β1 = g(x + 1) - g(x)

It is easy to understand that the coefficient β1 (the slope) represents the amount of change in the logit (g(x) for a change of one unit in the independent variable (x+1 versus x).

The interpretation of this change will depend upon the measurement scales used for the independent variable.

Dichotomous variable

Let's first assess what happens when the independent variable is a dichotomous variable (e.g. yes, no).

From the formula of the logit,

Ln [( P(y/x) / (1-P(y/x)] =  β0+ β1x1 

we had deducted that the exponential of the coefficient (e β1) is the ratio of the odds of disease among exposed to the odds of disease among unexposed (see Logistic model).

β1= Ln (Odds ratio)

e β1 = Odds ratio

In case of a dichotomous variable the odds ratio is the ratio of the odds for x = 1 to the odds of x = 0. The log of the odds ratio corresponds to the difference of the two logits with respectively x = 1 and x = 0.

The odds ratio gives us an idea of how much more likely (or less likely) it is for the outcome (e.g. disease) to occur among those with x = 1 (e.g. exposed) as compared to those with x = 0 (e.g. unexposed).

The confidence interval around the odds ratio can be computed as follows:

The following example shows the results of a logistic regression analysis of a study done during the investigation of a Salmonella outbreak in which the consumption of Tiramisu was suspected to be the vehicle of the epidemic. The data set includes 245 individuals including cases and controls.

Number of terms 2
Total Number of Observations 245
Rejected as Invalid 0
Number of valid Observations 245
Summary Statistics  Value DF p=value
Deviance 159,2489  243
Likelihood ratio test 180,3927 2  < 0.001 
Parameter Estimates  95% C.I 

 

Terms Coefficient Std.Error  p-value Odds Ratio Lower Upper
 %GM -2,9741  0,3875 < 0.001  0,0511  0,0239   0,1092
TIRA_   4,3116  0,4586 < 0.001 74,5578 30,3501 183,1579

In this output, β0 = - 2,9741and β1 = 4, 3116

The related OR is therefore equal to  e β1 = e 4,3116= 74,5578. It expresses that the odds of gastroenteritis are 74,5578 times higher among Tiramisu consumers than not.

The confidence interval, applying the above mentioned formula is [30,3501 - 183,1579].

Polytomous variable

The independent variable may have more than 2 categories. Let suppose that we want to assess the role of the amount of Tiramisu consumed in the outcome (gastroenteritis due to Salmonella) of the above example. In this example the independent variable has four categories (no consumption, small amount, medium and large). The categories are mutually exclusive. In the data set this variable was coded 0 for no consumption, 1 for small amount, 2 for medium and 3 for large amount. If an analysis is performed using this coding we obtain the following result.

Number of terms 2
Total Number of Observations 245
Rejected as Invalid 0
Number of valid Observations 245
Summary Statistics  Value DF p=value
Deviance 161,3985 243
Likelihood ratio test 178,2431 2  < 0.001 
Parameter Estimates  95% C.I 

 

Terms Coefficient Std.Error  p-value Odds Ratio Lower Upper
 %GM -2,5463 0,3048 < 0.001  0,0784  0,0431 0,1424
TIRA_   2,8479  0,3440  < 0.001 17,2518 8,7904 33,8580

In the above result, Tportion coded as 0,1,2 or 3 is interpreted as a continuous variable. Under the assumption that the logit is linear in the continuous variable Tportion, the value of the OR represents the amount by which the OR is multiplied by for each increase of one unit of Tportion.

In our example when we go from no exposure to small (0 to 1) the OR is 17,2518. When the exposure increases from 1 to 2 (small to medium amount) the OR is also 17,2518. This is meaning that moving from 0 to 2 the OR would be 17,2518 x 17,2518 = 297,62. Similarly moving from 0 to 3 (no consumption versus large amount) the OR would be 17,2518= 5134,56. This obviously does not represent the relation between dose and outcome. The values 0 to 3 have no numerical meaning. They are in fact a code for categories.

In such a situation we will create design variables (also called dummy variables or factor variables). The principle is that for n categories we need to create n-1 design variables.

Using the example of amount of Tiramisu the three design variables (D1, D2 and D3) take the following values. For no consumption D1, D2 and D3 will be assigned a 0 value. For small amount D1 = 1 and D2 and D3 = 0. For average consumption D1 = 0, D2 = 1 and D3 = 0. For large amount only D3 equals 1. This is summarised in the following table. Using a common reference group for coding design variables is the most frequent method. Most logistic regression software packages will generate design variables using this method.

Design (dummy) variables
Tiramisu consumption D1 D2 D3
None 0 0 0
Small 1 0 0
Medium 0 1 0
Large 0 0 1

Using the design variables in the above example, we obtain the following result.

    

Summary Statistics  Value DF p=value
Deviance 150,6160 241
Likelihood ratio test 189,0256 4  < 0.001 
Parameter Estimates  95% C.I 

 

Terms Coefficient Std.Error  p-value Odds Ratio Lower Upper
 %GM -2,9741 0,3875 < 0.001   0,0511  0,0239  0,1092
TPORTION2 ='1'  3,7518 0,4858  < 0.001 42,5966 16,4380 110,3828
TPORTION2 ='2'  5,3720 0,7168 < 0.001 215,2858 52,8285 877,3287
TPORTION2 ='3'  5,2767 1,1181  < 0.001 195,7142  21,8712 1751,3447

In the above table the odds ratio for eating a small amount (as compared to no consumption) is 42,5966. The odds ratio for eating a medium amount as compared to no consumption is 215,2858, for large amount it is 195,7142 (similar to the OR for medium portion). For the 3 categories the reference group is non consumers allowing therefore to compare odds ratios between categories of consumption.

Some important considerations on design variables merit to be noted:

  • All dummy variables should be considered as a single variable.
  • Each dummy variable corresponds to a degree of freedom (important in modelling).
  • Dummy variables can be created to indicate different levels of exposure (dose-response analysis).
  • Dummy variables can be created to indicate different levels of a quantitative variable (especially when doubting about linearity, see below).

Continuous independent variable

Some of the variables we will include in logistic models are continuous (e.g. age in years, weight in grams, height in cm, etc.). In such a case the interpretation of the coefficient will depend upon the unit chosen for the independent variable and the assumption that the logit is continuous in the dependent variable.

The logit can be expressed as:

g(x) =  β0+ β1x1

Here also the coefficient β1 gives the amount of change in the log odds (logit) for each unit of change in the independent variable.

The following table illustrates the relationship between age and occurrence of gastroenteritis due to Salmonella.       

Number of terms 2
Total Number of Observations 245
Rejected as Invalid

6

Number of valid Observations 239
Summary Statistics  Value DF p=value
Deviance 309,9103 237
Likelihood ratio test 21,4135 2  < 0.001 
Parameter Estimates  95% C.I 

 

Terms Coefficient Std.Error  p-value Odds Ratio Lower Upper
 %GM -0,6164 0,2914 0,0344 0,5399 0,3050  0,9556
AGE  0,0001 0,0100  0,9882 1,0001 0,9808 1,0199

The logit including age as a continuous variable is:

g(age) =  - 0,6164 + ( 0,0001 x AGE)

For one year increase of age the OR is multiplied by 1,0001 assuming linearity between age and log odds of gastoenteritis.  For 10 years increase the associated OR would be:

OR (10years) = exp (10 x 0,0001) =  2,72

In the above example age seems to increase slightly the risk of illness.

However if the logit is not linear then we should choose another way of analysis, e.g. creating categories of age (age groups) and develop the related design (dummy) variables or using other regression analysis techniques.

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