Creating The Data Set

# creating the data set
income <- c(6,8,10,13,15,20,25,30,35,40)
Ni <- c(40,50,60,80,100,70,65,50,40,25)
ni <- c(8,12,18,23,45,36,39,33,30,20)
dataTemp <- data.frame(cbind(income,Ni,ni))
# printing the data set
dataTemp
##    income  Ni ni
## 1       6  40  8
## 2       8  50 12
## 3      10  60 18
## 4      13  80 23
## 5      15 100 45
## 6      20  70 36
## 7      25  65 39
## 8      30  50 33
## 9      35  40 30
## 10     40  25 20

Calculating the Log Odds Ratio

Log Odds Ratio can be defined as

log(p/1-p)

Calculating the probability, Odds Ratio, Log Odds Ratio

# calculating probability
Pi <- (ni/Ni)
# calculating odd ratio
Odds <- Pi/(1-Pi)
# calculating Log Odds Ratio
LogOdds <- log(Odds)

Fitting the Logit Model

Regressing Income on LogOdds Ratio

# fitting the model
Logit <- lm(LogOdds ~ income,
                       data = dataTemp)

# summary of the model
summary(Logit)
## 
## Call:
## lm(formula = LogOdds ~ income, data = dataTemp)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.23736 -0.11364 -0.02464  0.09579  0.30791 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.720731   0.112400  -15.31 3.29e-07 ***
## income       0.080810   0.004862   16.62 1.74e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1728 on 8 degrees of freedom
## Multiple R-squared:  0.9719, Adjusted R-squared:  0.9683 
## F-statistic: 276.2 on 1 and 8 DF,  p-value: 1.736e-07