# Output of Logistic Regression Prediction

I have created a Logistic Regression using the following code:

full.model.f = lm(Ft_45 ~ ., LOG_D)
base.model.f = lm(Ft_45 ~ IP_util_E2pl_m02_flg)
step(base.model.f, scope=list(upper=full.model.f, lower=~1),
direction="forward", trace=FALSE)


I have then used the output to create a final model:

final.model.f = lm(Ft_45 ~ IP_util_E2pl_m02_flg + IP_util_E2_m02_flg +
AE_NumVisit1_flg + OP_NumVisit1_m01_flg + IP_TotLoS_m02 +
Ft1_45 + IP_util_E1_m05_flg + IP_TotPrNonElecLoS_m02 +
IP_util_E2pl_m03_flg + LTC_coding + OP_NumVisit0105_m03_flg +
OP_NumVisit11pl_m03_flg + AE_ArrAmb_m02_flg)


Then I have predicted the outcomes for a different set of data using the predict function:

log.pred.f.v <- predict(final.model.f, newdata=LOG_V)


I have been able to use establish a pleasing ROC curve and created a table to establish the sensitivity and specificity which gives me responses I would expect.

However What I am trying to do is establish for each row of data what the probability is of Ft_45 being 1. If I look at the output of log.pred.f.v I get, for example,:

1 -0.171739593
2 -0.049905948
3 0.141146419
4 0.11615669
5 0.07342591
6 0.093054334
7 0.957164383
8 0.098415639
.
.
.
104 0.196368229
105 1.045208447
106 1.05499112


As I only have a tentative grasp on what I am doing I am struggling to understand how to interpret the negative and higher that 1 values as I would expect a probability to be between 0 and 1.

So my question is am I just missing a step where I need to transform the output or have I gone completely wrong.

First, it looks like you built a regular linear regression model, not a logistic regression model. To build a logistic regression model, you need to use glm() with  family="binomial" , not lm().

Suppose you build the following logistic regression model using independent variables $x_1, x_2$, and $x_3$ to predict the probability of event $y$:

logit <- glm(y~x1+x2+x3,family="binomial")


This model has regression coefficients $\beta_0, \beta_1, \beta_2$ and $\beta_3$.

If you then do predict(logit), R will calculate and return b0 + b1*x1 + b2*x2 + b3*x3.

Recall that your logistic regression equation is $y = log(\frac{p}{1-p}) = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3$.

So, to get the probabilities that you want, you need to solve this equation for $p$.

In R, you can do something like this:

pred <- predict(logit,newdata=data) #gives you b0 + b1x1 + b2x2 + b3x3
probs <- exp(pred)/(1+exp(pred)) #gives you probability that y=1 for each observation