In the least square method, the regression model is established in such a way that the sum of the squares of the vertical distances of different points (residuals) from the regression line is minimized. When the relationship between the variables is not linear (one has a non-linear regression model), one may

1. try to transform the data to linearize the relationship,
2. fit polynomial or complex spline model to the data, or
3. fit a non-linear regression to the data.

In the non-linear regression model, a function is specified by a set of parameters to fit the data. The non-linear least squares approach is used to estimate such parameters. In R, the nls() is used to approximate the non-linear function using a linear one and iteratively try to find the best parameter values.

Some frequently used non-linear regression models are listed in the Table below.

Let fit Michaelis-Menten non-linear function to the data given below.

x <- seq(1, 10, 1)
y <- c(3.7, 7.1, 11.9, 19, 27, 38.5, 51, 67.7, 85, 102)

nls_model <- nls(y ~ a * x/(1 + b * x), start = list(a = 1, b = 1))

summary(nls_model)

#### Output
Formula: y ~ a * x/(1 + b * x)

Parameters:
   Estimate Std. Error t value Pr(>|t|)    
a  4.107257   0.226711   18.12 8.85e-08 ***
b -0.060900   0.002708  -22.49 1.62e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.805 on 8 degrees of freedom

Number of iterations to convergence: 11 
Achieved convergence tolerance: 6.354e-06

Let plot the non-linear predicted values from 10 data points of newly generated x-values

new.data <- data.frame(x = seq(min(x), max(x), len = 10))

plot(x, y)

lines(new.data$x, predict(nls_model, newdata = new.data) )

The sum of squared residuals and the confidence interval of the chosen values of the coefficient can be obtained by issuing the commands,

sum(resid(nls_model)^2) 
# or 
print(sum(resid(nls_model)^2))

confint(nls_model) 
# or 
print(confint(nls_model))

Note that the formula for nls() does not use special coding in linear terms, factors, interactions, etc. The right-hand side in the expression of nls() computes the expected value to the left-hand side. The start argument contains the list of starting values of the parameter used in the expression and is varied by the algorithm.

In logistic regression models, the response variable ($y$) is of categorical (binary, dichotomous) values such as 1 or 0 (TRUE/ FALSE). It measures the probability of a binary response variable based on mathematical equation relating the values of response variable with the predictor(s). The built-in function glm() can be used to perform logistic regression analysis.

The odds are used in logistic regression. If $p$ is the probability of success, the odds of in favour of success are, $\frac{p}{q}=\frac{p}{1-p}$.

Note that probability can be converted to odds and odds can also be converted to probability. However, unlike probability, odds can exceed 1. For example, if the probability of an event is 0.25, the odds in the favour of that event are $\frac{0.25}{0.75}=0.33$, but the odds against the event are $\frac{0.75}{0.25}=3$.

In built-in dataset (“mtcars“), the column (am) describes the transmission mode (automatic or manual) which is of binary value (0 or 1). The logistic regression models between the column (response variable) am and other columns (regressors) namely hp, wt, and cyl can be performed as given are:

Logistic regression with one dichotomous predictor

logmodel1 <- glm(am ~ vs, family = "binomial")
summary(logmodel1)

Logistic regression with one continuous predictor

logmodel2 <- glm(am ~ wt, family = "binomial")
summary(logmodel2)

Logistic regression with multiple predictors

logmodel3 <- glm(am ~ cyl + hp + wt, family = "binomial")
summary(logmodel3)
plot(logmodel3)

Note: in logistic regression model, both dichotomous and continuous variables can be used as predictor.

In R, the coefficients returned by logistic regression are a logit, or the log of the odds. To convert logits to odds ratio exponentiate it and to convert logits to probability use $\frac{e^\beta}{1-e^\beta}$. For example,

logmodel1 <- glm(am ~ vs, family = "binomial", data = mtcars)
logit_coef <- logmodel1$coef

exp(logmodel1$coef)
exp(logit_coef)/(1 + exp(logmodel1$coef))