Lambda and Gamma as Defined in Sociology

Lambda and gamma are two measures of association that are commonly used in social science statistics and research. Lambda is a measure of association used for nominal variables while gamma is used for ordinal variables.

Lambda

Lambda is defined as an asymmetrical measure of association that is suitable for use with nominal variables. It may range from 0.0 to 1.0. Lambda provides us with an indication of the strength of the relationship between independent and dependent variables. As an asymmetrical measure of association, lambda’s value may vary depending on which variable is considered the dependent variable and which variables are considered the independent variable.

To calculate lambda, you need two numbers: E1 and E2. E1 is the error of prediction made when the independent variable is ignored. To find E1, you first need to find the mode of the dependent variable and subtract its frequency from N. E1 = N – Modal frequency.

E2 is the errors made when the prediction is based on the independent variable. To find E2, you first need to find the modal frequency for each category of the independent variables, subtract it from the category total to find the number of errors, then add up all the errors.

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The formula for calculating lambda is: Lambda = (E1 – E2) / E1.

Lambda may range in value from 0.0 to 1.0. Zero indicates that there is nothing to be gained by using the independent variable to predict the dependent variable. In other words, the independent variable does not, in any way, predict the dependent variable. A lambda of 1.0 indicates that the independent variable is a perfect predictor of the dependent variable. That is, by using the independent variable as a predictor, we can predict the dependent variable without any error.

Gamma

Gamma is defined as a symmetrical measure of association suitable for use with ordinal variable or with dichotomous nominal variables. It can vary from 0.0 to +/- 1.0 and provides us with an indication of the strength of the relationship between two variables. Whereas lambda is an asymmetrical measure of association, gamma is a symmetrical measure of association. This means that the value of gamma will be the same regardless of which variable is considered the dependent variable and which variable is considered the independent variable.

Gamma is calculated using the following formula:

Gamma = (Ns - Nd)/(Ns + Nd)

The direction of the relationship between ordinal variables can either be positive or negative. With a positive relationship, if one person ranked higher than another on one variable, he or she would also rank above the other person on the second variable. This is called same order ranking, which is labeled with an Ns, shown in the formula above. With a negative relationship, if one person is ranked above another on one variable, he or she would rank below the other person on the second variable. This is called an inverse order pair and is labeled as Nd, shown in the formula above.

To calculate gamma, you first need to count the number of same order pairs (Ns) and the number of inverse order pairs (Nd). These can be obtained from a bivariate table (also known as a frequency table or crosstabulation table). Once these are counted, the calculation of gamma is straightforward.

A gamma of 0.0 indicates that there is no relationship between the two variables and nothing is to be gained by using the independent variable to predict the dependent variable. A gamma of 1.0 indicates that the relationship between the variables is positive and the dependent variable can be predicted by the independent variable without any error. When gamma is -1.0, this means that the relationship is negative and that the independent variable can perfectly predict the dependent variable with no error.

References

• _{Frankfort-Nachmias, C. & Leon-Guerrero, A. (2006). Social Statistics for a Diverse Society. Thousand Oaks, CA: Pine Forge Press.}