[sphweb.bumc.bu.edu/otlt/MPH-Modules/PH717-QuantCore/PH717_MultipleVariableRegression/PH717_MultipleVariableRegression6.html](https://sphweb.bumc.bu.edu/otlt/MPH-Modules/PH717-QuantCore/PH717_MultipleVariableRegression/PH717_MultipleVariableRegression6.html) 

Multiple Linear Regression in R

```R 
#confint() provides information on confidence intervals for the coefficient estimates
#This can be used with both simple and multiple linear regression

confint(lm.fit)
```
Below is an example of the output information you can get by calling confint(lm.fit):

(from Pg. 111 of Introduction to Statistical Learning with Applications in R).

University of Michigan - Ann Arbor

``` R
### start with assigning lm.fit to the model:
lm.fit = lm(XVAL~YVAL, data = datavar)
#1.  lm.fit provides “basic” information *such as* Coefficients
lm.fit
```
Below is an example of the type of output information you can get calling lm.fit (from Pg. 111 of Introduction to Statistical Learning with Applications in R).


lm.fit: Output Basic Statistical Information in R 

Multiple Linear Regression is a linear regression that uses **two or more independent variables** to predict the outcome of a response variable. It is more effective and practical to predict or estimate dependent variables by the optimal combination of multiple independent variables than only one independent variable. 
$$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + ... + \beta_p x_{ip} + \epsilon$$

Multiple Linear Regression

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

``` R
#summary(lm.fit) provides “detailed” information on the model including:
#1.) R^2 statistic
#2.) F-statistic
#3.) p-values
#4.) standard errors for coefficients for the model
summary(lm.fit)
```
Below is an example of the type of output information you can get calling **summary(lm.fit)**

(from Pg. 111 of Introduction to Statistical Learning with Applications in R).


summary(lm.fit): Output Detailed Statistical Information in R

- predict() returns y-hat (estimate) based on your training model. 
- This method produces the confidence and prediction intervals for the prediction of YVAL given a value of the XVAL. In other words, for each training set, we have corresponding x and y values. 
- Now we have a new data set and we are passing a vector with one column from that new data and we want to predict the prediction/confidence intervals for that new data value based on the training data model.
```R
predict(lm.fit, data.frame(XVAL=(c(new_data_row_value1, new_data_row_value2), interval = “”)
#Example: 
predict(lm.fit, data.frame(XVAL=(c(5, 10,15)), interval = “prediction”)
	#Above, data.frame is included as an optional data frame that looks for variables to predict. Without it, fitted values are used.
	#interval = “confidence” , “none”, “prediction” are all options
        ```

predict(): Predict New Data from the (Simple) Linear Model in R

confint() in R with linear regression

https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/lm.fit

lm.fit Documentation in R

More often than not, there are predictors in our data set that are not associated with the response. Because of this it is useful to narrow down our model to include only a subset of predictors.
- Forward Selection
- Backward Selection
- Mixed Selection


Methods for deciding important variable

Here is an example of multiple linear regression using R:
```r
library(MASS) #  MASS package is a very large collection of data sets and functions
library(ISLR) # ISLR package includes the data sets associated with this book
lm.fit=lm(medv∼lstat+age,data=Boston)# medv as a response, lstat and age as two predictors
summary(lm.fit) # gives us p-values and standard errors for the coeﬃcients, as well as the R2 statistic and F-statistic for the model

#The syntax lm(y∼x1+x2+x3) is used to ﬁt a model with three predictors, #x1, x2, andx3. The summary() function now outputs the regression #coeﬃcients for all the predictors. 

#We can also make a regression using all predictors with the following code:
lm.fit=lm(medv∼.,data=Boston) 
```

Learn Before

Related

Learn After