CourseKata - 8.2 Fitting a Regression Model

Introduction to Statistics: A Modeling Approach

Fitting a Regression Model

Using `lm()` to Fit the Height Model to TinyFingers

Now you can begin to see the power you’ve been granted by the General Linear Model! Fitting—or estimating the parameters—of the regression model is accomplished the same way as estimating the parameters of the grouping model. It’s all done using the lm() function in R.

The lm() function is smart enough to know that if the explanatory variable is quantitative, the model to estimate is the regression model. If the explanatory variable is categorical (e.g., defined as a factor in R), lm() will fit a group model.

Modify the code below to fit the regression model using Height as the explanatory variable to predict Thumb length in the TinyFingers data.

                
                require(ggformula)
                require(mosaic)
                require(Lock5Data)
                require(Lock5withR)
                require(okcupiddata)
                Fingers <- read.csv(file="https://raw.githubusercontent.com/UCLATALL/intro-stats-modeling/master/datasets/fingers.csv", header=TRUE, sep=",")
                Fingers <- data.frame(Fingers)
                #set up tiny data set
                Thumb <- c(56, 60, 61, 63, 64, 68)
                Sex <- c("female","female","female","male","male","male")
                TinyFingers <- data.frame(Sex, Thumb)                    
                TinyFingers$Sex <- as.factor(TinyFingers$Sex)
                TinyFingers$Height = c(62, 66, 67, 63, 68, 71)
                TinyFingers$Height2Group = ntile(TinyFingers$Height, 2)
                TinyFingers$Height3Group = ntile(TinyFingers$Height, 3)
                TinyFingers$Height2Group = recode(TinyFingers$Height2Group, '1' = "1-Short", '2' = "2-Tall")
                TinyFingers$Height3Group = recode(TinyFingers$Height3Group, '1' = "1-Short", '2' = "2-Medium", '3' = "3-Tall")


                # modify this to fit the model
                TinyHeight.model <- lm()

                # this prints the best fitting estimates
                TinyHeight.model


                # modify this to fit the model
                TinyHeight.model <- lm(Thumb ~ Height, data = TinyFingers)

                # this prints the best fitting estimates
                TinyHeight.model


                test_object("TinyHeight.model")
                test_output_contains("TinyHeight.model")
                test_error()
                success_msg("Keep up the great work!")

Model Thumb length as a function of Height

DataCamp: ch8-1

source: Ch8_TinyFingers-output

L_Ch8_Fitting_1

Fitting a Regression Model By Accident When You Don’t Want One

Although R is pretty smart about knowing which model to fit, it won’t always do the right thing. If you code the grouping variable with the character strings “short” and “tall,” R will make the right decision because it knows the variable must be categorical. But if you code a grouping variable as 1 and 2, and you forget to make it a factor, R may get confused and fit the model as though the explanatory variable is quantitative.

For example, we’ve added a new variable to our TinyFingers data called GroupNum. Here is what the data look like.

source: Ch8_GroupNum

If you take a look at the variables Height2Group and GroupNum, they have the same information. Students 1, 2, and 4 are in one group and students 3, 5, and 6 are in another group. If we fit a model with Height2Group (and called it the Height2Group.model) or GroupNum (and called it the GroupNum.model), we would expect the same estimates. Let’s try it.

                
                require(mosaic)
                require(ggformula)
                #set up tiny data set
                Thumb <- c(56, 60, 61, 63, 64, 68)
                Sex <- c("female","female","female","male","male","male")
                TinyFingers <- data.frame(Sex, Thumb)
                TinyFingers$Sex <- as.factor(TinyFingers$Sex)
                TinyFingers$Height = c(62, 66, 67, 63, 68, 71)
                TinyFingers$Height2Group = ntile(TinyFingers$Height, 2)
                TinyFingers$Height3Group = ntile(TinyFingers$Height, 3)
                TinyFingers$Height2Group = recode(TinyFingers$Height2Group, '1' = "1-Short", '2' = "2-Tall")
                TinyFingers$Height3Group = recode(TinyFingers$Height3Group, '1' = "1-Short", '2' = "2-Medium", '3' = "3-Tall")
                
                TinyFingers$GroupNum <- ntile(TinyFingers$Height, 2)


                # fit a model of Thumb length based on Height2Group
                Height2Group.model <- lm()
                                 
                # fit a model of Thumb length based on GroupNum 
                GroupNum.model <- lm()
                
                # this prints the parameter estimates from the two models 
                Height2Group.model
                GroupNum.model


                # fit a model of Thumb length based on Height2Group
                Height2Group.model <- lm(Thumb ~ Height2Group, data = TinyFingers)
                                 
                # fit a model of Thumb length based on GroupNum 
                GroupNum.model <- lm(Thumb ~ GroupNum, data = TinyFingers)
                
                # this prints the parameter estimates from the two models 
                Height2Group.model
                GroupNum.model


                test_object("Height2Group.model")
                test_object("GroupNum.model")
                test_output_contains("Height2Group.model")
                test_output_contains("GroupNum.model")
                success_msg("Great work!")

DataCamp: ch8-2

source: Ch8_TinyFingers-output2

L_Ch8_Fitting_2

Because Height2Group is a factor (i.e., a categorical variable), lm() fits a group model. But for GroupNum, lm() thinks the 1 or 2 coding refers to a quantitative variable because we did not tell R that it was a factor. So it fits a regression line instead of a group model. If it does that, the meaning of the estimates will not be what you expect for the group model.

The slope will be accurate, because it will tell you the increment in thumb length between people coded as 2 vs. those coded as 1. But the \(b_{0}\) estimate will be the y-intercept—i.e., the predicted thumb length when \(X_{i}\) equals 0. This makes no sense when there are only two groups and they are coded 1 and 2. This is an accidental regression model.

L_Ch8_Fitting_3

Try it here by recoding GroupNum as 0 and 1. See if the results fit your expectations.

                
                require(mosaic)
                require(ggformula)
                #set up tiny data set
                Thumb <- c(56, 60, 61, 63, 64, 68)
                Sex <- c("female","female","female","male","male","male")
                TinyFingers <- data.frame(Sex, Thumb)
                TinyFingers$Sex <- as.factor(TinyFingers$Sex)
                TinyFingers$Height = c(62, 66, 67, 63, 68, 71)
                TinyFingers$Height2Group = ntile(TinyFingers$Height, 2)
                TinyFingers$Height3Group = ntile(TinyFingers$Height, 3)
                TinyFingers$Height2Group = recode(TinyFingers$Height2Group, '1' = "1-Short", '2' = "2-Tall")
                TinyFingers$Height3Group = recode(TinyFingers$Height3Group, '1' = "1-Short", '2' = "2-Medium", '3' = "3-Tall")
                TinyFingers$GroupNum <- ntile(TinyFingers$Height, 2)
                TinyFingers$Group01 <- ntile(TinyFingers$Height, 2)
                Height2Group.model <- lm(Thumb ~ Height2Group, data = TinyFingers)


               # recode GroupNum from 1 and 2 to 0 and 1
                TinyFingers$GroupNum <- recode()
                                            
                # This will fit an accidental regression model 
                GroupNum.model <- lm(Thumb ~ GroupNum, data = TinyFingers) 
                GroupNum.model

               
                # recode GroupNum from 1 and 2 to 0 and 1
                TinyFingers$GroupNum <- recode(TinyFingers$GroupNum, "1" = 0, "2" = 1)
                                            
                # This will fit an accidental regression model 
                GroupNum.model <- lm(Thumb ~ GroupNum, data = TinyFingers) 
                GroupNum.model


                test_data_frame("TinyFingers") 
                test_object("GroupNum.model")
                test_output_contains("GroupNum.model")
                success_msg("Great work!")

DataCamp: ch8-2a

Fitting the Height Model to the Full Fingers Data Set

Now that you have looked in detail at the tiny set of data, fit the height model to the full Fingers data frame, and save the model in an R object called Height.model.

                
                require(mosaic)
                require(ggformula)
                Fingers <- read.csv(file="https://raw.githubusercontent.com/UCLATALL/intro-stats-modeling/master/datasets/fingers.csv", header=TRUE, sep=",")
                # this is for measurement section
                #Fingers <- arrange(Fingers, desc(Sex))
                #Fingers$FamilyMembers[1] <- 2
                #Fingers$Height[1] <- 62
                #Fingers$Sex <- recode(Fingers$Sex, '1' = "female", '2' = "male")

                Fingers <- data.frame(Fingers)
                # clean up str

                Fingers$Sex <- as.factor(Fingers$Sex)
                Fingers$RaceEthnic <- as.numeric(Fingers$RaceEthnic)
                Fingers$SSLast <- as.numeric(Fingers$SSLast)
                Fingers$Year <- as.numeric(Fingers$Year)
                Fingers$Job <- as.numeric(Fingers$Job)
                Fingers$MathAnxious <- as.numeric(Fingers$MathAnxious)
                Fingers$Interest <- as.numeric(Fingers$Interest)
                Fingers$GradePredict <- as.numeric(Fingers$GradePredict)
                Fingers$Thumb <- as.numeric(Fingers$Thumb)
                Fingers$Index <- as.numeric(Fingers$Index)
                Fingers$Middle <- as.numeric(Fingers$Middle)
                Fingers$Ring <- as.numeric(Fingers$Ring)
                Fingers$Pinkie <- as.numeric(Fingers$Pinkie)
                Fingers$Height <- as.numeric(Fingers$Height)
                Fingers$Weight <- as.numeric(Fingers$Weight)

                Fingers <- filter(Fingers, Thumb >= 33 & Thumb <= 100)
                set.seed(2)


                # modify this to fit the Height model of Thumb for the Fingers data
                Height.model <- 
  
                # this prints best estimates
                Height.model


                Height.model <- lm(Thumb ~ Height, data = Fingers)
                Height.model


                test_object("Height.model")
                test_output_contains("Height.model")
                success_msg("Awesome job!")

Model Thumb as a function of Height

DataCamp: ch8-3

source: Ch8_Fingers-output

L_Ch8_Fitting_4

Here is the code to make a scatter plot to show the relationship between Height (on the x-axis) and Thumb (on the y-axis) for TinyFingers. Note that the code also overlays the best-fitting regression line on the scatter plot. Edit the code to make this scatter plot for the full Fingers data set.

                
                               require(mosaic)
                require(ggformula)
                Fingers <- read.csv(file="https://raw.githubusercontent.com/UCLATALL/intro-stats-modeling/master/datasets/fingers.csv", header=TRUE, sep=",")
                # this is for measurement section
                #Fingers <- arrange(Fingers, desc(Sex))
                #Fingers$FamilyMembers[1] <- 2
                #Fingers$Height[1] <- 62
                #Fingers$Sex <- recode(Fingers$Sex, '1' = "female", '2' = "male")

                Fingers <- data.frame(Fingers)
                # clean up str

                Fingers$Sex <- as.factor(Fingers$Sex)
                Fingers$RaceEthnic <- as.numeric(Fingers$RaceEthnic)
                Fingers$SSLast <- as.numeric(Fingers$SSLast)
                Fingers$Year <- as.numeric(Fingers$Year)
                Fingers$Job <- as.numeric(Fingers$Job)
                Fingers$MathAnxious <- as.numeric(Fingers$MathAnxious)
                Fingers$Interest <- as.numeric(Fingers$Interest)
                Fingers$GradePredict <- as.numeric(Fingers$GradePredict)
                Fingers$Thumb <- as.numeric(Fingers$Thumb)
                Fingers$Index <- as.numeric(Fingers$Index)
                Fingers$Middle <- as.numeric(Fingers$Middle)
                Fingers$Ring <- as.numeric(Fingers$Ring)
                Fingers$Pinkie <- as.numeric(Fingers$Pinkie)
                Fingers$Height <- as.numeric(Fingers$Height)
                Fingers$Weight <- as.numeric(Fingers$Weight)

                Fingers <- filter(Fingers, Thumb >= 33 & Thumb <= 100)
                set.seed(2)


                # edit this code to create a scatter plot for teh full Fingers data
                gf_point(Thumb ~ Height, data = TinyFingers, size = 4) %>%
                    gf_lm(color = "orange")


                 gf_point(Thumb ~ Height, data = Fingers ) %>%
                    gf_lm(Thumb ~ Height, data = Fingers, color = "orange")


                test_function("gf_point", args = "data")
                test_function("gf_lm", args = "data")
                test_error()
                success_msg("You R an R wizard!")

DataCamp: ch8-4

source: Ch8_Thumb-height2

8.1 The Regression Line as a Model 8.3 Using the Regression Model to Make Predictions

Course Outline

Introduction to Statistics: A Modeling Approach

Fitting a Regression Model

Using `lm()` to Fit the Height Model to TinyFingers

Fitting a Regression Model By Accident When You Don’t Want One

Fitting the Height Model to the Full Fingers Data Set

Responses

list Introduction to Statistics: A Modeling Approach

Fitting a Regression Model

Using lm() to Fit the Height Model to TinyFingers

Fitting a Regression Model By Accident When You Don’t Want One

Fitting the Height Model to the Full Fingers Data Set

Introduction to Statistics: A Modeling Approach

Using `lm()` to Fit the Height Model to TinyFingers