Generalized-Linear-Models.md

@@ -142,6 +142,28 @@ In this linear predictor, we can also use transformation of the predictors as we
     onto the linear predictor. This function should be chosen in such a way to ensure that 
     the predicted means are in the permissible range. 
 
+    <img src="https://latex.codecogs.com/svg.latex?g(\mu)=\eta" title="g(\mu)=\eta" />
+
+    This link function should be monotonic and differentiable. As a result, the mean of the 
+    response variable would be linked to the predictor variables as: 
+
+    <img src="https://latex.codecogs.com/svg.latex?\mu&space;=&space;g^{-1}(\eta)=g^{-1}(\beta_0&plus;\beta_1x_1&plus;...&plus;\beta_Qx_Q)" title="\mu = g^{-1}(\eta)=g^{-1}(\beta_0+\beta_1x_1+...+\beta_Qx_Q)" />
+
+    The important thing here is that this link relates the mean of the response to the 
+    linear predictor and this is different from transforming the response variable. 
+
+    An important consideration in choosing a link function is whether the selected link 
+    will yield predicted values that are permissible. As an example, a common link for non- 
+    negative count data or reaction times would be the natural logarithm. When the data are 
+    not bounded, an identical function or a reciprocal function can be used. For the data 
+    that are bounded between 0 and 1 a cumulative distribution function can be chosen (such 
+    as logistic (logit function), normal (probit function), or Gumbal (log-log function) 
+    distributions). 
+
+    For the distributions of the natural exponential family, special link functions exist which are called *Canonical Link Functions*. By using the canonical link functions, we assure that the natural parameter of the distribution <img src="https://latex.codecogs.com/svg.latex?\theta" title="\theta" /> equals the linear predictor <img src="https://latex.codecogs.com/svg.latex?\eta" title="\eta" />. These functions are a good initial choice to start our modeling procedure but in some cases they may not be the best option. 
+
+
+