Units of the parameters

The parameter β0 has the same units as the Y values (the outcome variable).

The other best-fit parameters have the units of the Y variable divided by the units of the corresponding X variable.

Consider this example model again:

Blood pressure ~ Intercept + Age + Weight + Gender

As well as its mathematical form:

Blood pressure = β0 + β1*Age + β2*Weight + β3*Gender[Male]

For this model, β0 is the estimate of the model intercept, and is expressed in units of the Y variable, which is mmHg. It is the predicted value of Y when all continuous predictor variables equal zero and all categorical predictor variables are set to their reference level. For this example (and many others) this is a bit silly, as it would be the average blood pressure of women (assuming the reference level of Gender is “Female”) with age=0 and weight=0! In this case, it’s better just to think of it as a constant in the model. However, this constant is very important when it comes to predicting values of the outcome variable using different (non-zero) values for the predictor variables via interpolation.

If  blood pressure is measured in mmHg and age is measured in years, the variable β1 will have units of mmHg/year. It is the amount by which blood pressure increases, on average, for every year increase in age, after correcting for differences in gender and weight.

If weight is measured in kg, then β2 has units of mmHg/kg. It is the average amount by which blood pressure increases for every kg increase in weight, adjusting for differences in age and gender.

Gender is a categorical predictor variable with no units. Prism automatically codes categorical variables for use in regression, and will also choose a default reference level to use when performing the dummy coding (this default reference level can be changed using the Reference Level tab of the regression dialog). In this case, the level “Female” was the reference level, so females were coded as 0 and males as 1. Therefore, β3 has units of mmHg. It is the average difference in blood pressure between men and women, after taking into account differences in age and weight.

In the results that Prism generates, you can tell which level of a categorical variable is the reference level as it will be the only level that is not given a parameter estimate.

Standard errors, t statistics, and confidence intervals

The only way you could really know the best-fit values of the parameters in the model would be to collect an infinite amount of data. Since you can't do this, the best-fit values reported by Prism are influenced, in part, by random variability in picking subjects. Prism reports this uncertainty as a 95% confidence interval for each parameter. These take into account the number of subjects in your study, as well as the scatter of your data from the predictions of the model. If the assumptions of the analysis are true, you can be 95% sure the true best-fit value of the parameter lies within that confidence interval.

Prism also presents the standard error of each parameter in the model. These are hard to interpret, but are used to compute the t statistics and 95% confidence intervals for each coefficient. Prism shows them so that its results can be compared to those of other programs.

Prism also reports the absolute value of the t statistic for each parameter, which is computed as the parameter value divided by its standard error. For Poisson regression, Prism reports the absolute value of z instead.

The 95% confidence interval is the standard error of the parameter multiplied by a critical t ratio, determined  a constant based on sample size, and adding or subtracting this product from the estimate.

Parameter estimates for levels of categorical predictor variables

When a categorical predictor variable is included in a regression model, Prism automatically encodes this variable using “dummy coding”. This process generates (behind the scenes) a number of new variables equal to the number of levels of the original categorical variable minus one. In other words, if a categorical predictor variable had 5 unique levels (A, B, C, D, and E, for example), dummy coding would generate 4 new variables. If a categorical predictor variable only had two unique levels (Male and Female, for example), dummy coding would generate only one new variable. In this way, every level of a categorical predictor variable except for one gets a new variable that is used in the regression analysis. Additionally, a beta coefficient is estimated for each of these new variables. The level that does not get a new variable or regression coefficient is the reference level.

These parameter estimates (coefficients) are listed for each level of the categorical variable in the Tabular results of the regression analysis. Knowing which coefficient belongs with each individual level is relatively simple, as the variable name will be in the form “Variable[Level]”. For example, in the variable “Gender”, the coefficient β3 will correspond to Gender[Female], and represents how much higher (or lower) the average blood pressure for women is compared to men after taking into account differences in age and weight.