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.Rmd file as a template.lm)For this exercise we will use the data stored in nutrition-2018.csv. It contains the nutritional values per serving size for a large variety of foods as calculated by the USDA in 2018. It is a cleaned version totaling 5956 observations and is current as of April 2018.
The variables in the dataset are:
IDDesc - short description of foodWater - in gramsCaloriesProtein - in gramsFat - in gramsCarbs - carbohydrates, in gramsFiber - in gramsSugar - in gramsCalcium - in milligramsPotassium - in milligramsSodium - in milligramsVitaminC - vitamin C, in milligramsChol - cholesterol, in milligramsPortion - description of standard serving size used in analysis(a) Fit the following multiple linear regression model in R. Use Calories as the response and Fat, Sugar, and Sodium as predictors.
\[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \epsilon_i. \]
Here,
Calories.Fat.Sugar.Sodium.Use an \(F\)-test to test the significance of the regression. Report the following:
When reporting these, you should explicitly state them in your document, not assume that a reader will find and interpret them from a large block of R output.
(b) Output only the estimated regression coefficients. Interpret all \(\hat{\beta}_j\) coefficients in the context of the problem.
(c) Use your model to predict the number of Calories in a Filet-O-Fish. According to McDonald’s publicized nutrition facts, the Filet-O-Fish contains 18g of fat, 5g of sugar, and 580mg of sodium.
(d) Calculate the standard deviation, \(s_y\), for the observed values in the Calories variable. Report the value of \(s_e\) from your multiple regression model. Interpret both estimates in the context of this problem.
(e) Report the value of \(R^2\) for the model. Interpret its meaning in the context of the problem.
(f) Calculate a 90% confidence interval for \(\beta_2\). Give an interpretation of the interval in the context of the problem.
(g) Calculate a 95% confidence interval for \(\beta_0\). Give an interpretation of the interval in the context of the problem.
(h) Use a 99% confidence interval to estimate the mean Calorie content of a food with 15g of fat, 0g of sugar, and 260mg of sodium, which is true of a medium order of McDonald’s french fries. Interpret the interval in context.
(i) Use a 99% prediction interval to predict the Calorie content of a Crunchy Taco Supreme, which has 11g of fat, 2g of sugar, and 340mg of sodium according to Taco Bell’s publicized nutrition information. Interpret the interval in context.
lm for Multiple Regression)For this exercise we will use the data stored in goalies17.csv. It contains career data for goaltenders in the National Hockey League during the first 100 years of the league from the 1917-1918 season to the 2016-2017 season. It holds the 750 individuals who played at least one game as goalie over this timeframe. The variables in the dataset are:
Player - Player’s Name (those followed by * are in the Hall of Fame as of 2017)First - First year with game recorded as goalieLast - Last year with game recorded as goalieActive - Number of seasons active in the NHLGP - Games PlayedGS - Games StartedW - WinsL - Losses (in regulation)TOL - Ties and Overtime LossesGA - Goals AgainstSA - Shots AgainstSV - SavesSV_PCT - Save PercentageGAA - Goals Against AverageSO - ShutoutsPIM - Penalties in MinutesMIN - MinutesFor this exercise we will consider three models, each with Wins as the response. The predictors for these models are:
After reading in the data but prior to any modeling, you should clean the data set for this exercise by removing the following variables: Player, GS, L, TOL, SV_PCT, and GAA.
(a) Use an \(F\)-test to compares Models 1 and 2. Report the following:
(b) Use an \(F\)-test to compare Model 3 to your preferred model from part (a). Report the following:
(c) Use a \(t\)-test to test \(H_0: \beta_{\texttt{SV}} = 0 \ \text{vs} \ H_1: \beta_{\texttt{SV}} \neq 0\) for the model you preferred in part (b). Report the following:
lm)For this exercise we will once again use the Ozone data from the mlbench package. The goal of this exercise is to fit a model with ozone as the response and the remaining variables as predictors.
data(Ozone, package = "mlbench")
Ozone = Ozone[, c(4, 6, 7, 8)]
colnames(Ozone) = c("ozone", "wind", "humidity", "temp")
Ozone = Ozone[complete.cases(Ozone), ](a) Obtain the estimated regression coefficients without the use of lm() or any other built-in functions for regression. That is, you should use only matrix operations. Store the results in a vector beta_hat_no_lm. To ensure this is a vector, you may need to use as.vector(). Return this vector as well as the results of sum(beta_hat_no_lm ^ 2).
(b) Obtain the estimated regression coefficients with the use of lm(). Store the results in a vector beta_hat_lm. To ensure this is a vector, you may need to use as.vector(). Return this vector as well as the results of sum(beta_hat_lm ^ 2).
(c) Use the all.equal() function to verify that the results are the same. You may need to remove the names of one of the vectors. The as.vector() function will do this as a side effect, or you can directly use unname().
(d) Calculate \(s_e\) without the use of lm(). That is, continue with your results from (a) and perform additional matrix operations to obtain the result. Output this result. Also, verify that this result is the same as the result obtained from lm().
(e) Calculate \(R^2\) without the use of lm(). That is, continue with your results from (a) and (d), and perform additional operations to obtain the result. Output this result. Also, verify that this result is the same as the result obtained from lm().
For this exercise use the Auto dataset from the ISLR package. Use ?Auto to learn about the dataset. The goal of this exercise is to find a model that is useful for predicting the response mpg. We remove the name variable as it is not useful for this analysis. (Also, this is an easier to load version of data from the textbook.)
# load required package, remove "name" variable
library(ISLR)## Warning: package 'ISLR' was built under R version 4.3.3Auto = subset(Auto, select = -c(name))
dim(Auto)## [1] 392 8When evaluating a model for prediction, we often look at RMSE. However, if we both fit the model with all the data as well as evaluate RMSE using all the data, we’re essentially cheating. We’d like to use RMSE as a measure of how well the model will predict on unseen data. If you haven’t already noticed, the way we had been using RMSE resulted in RMSE decreasing as models became larger.
To correct for this, we will only use a portion of the data to fit the model, and then we will use leftover data to evaluate the model. We will call these datasets train (for fitting) and test (for evaluating). The definition of RMSE will stay the same
\[ \text{RMSE}(\text{model, data}) = \sqrt{\frac{1}{n} \sum_{i = 1}^{n}(y_i - \hat{y}_i)^2} \]
where
However, we will now evaluate it on both the train set and the test set separately. So each model you fit will have a train RMSE and a test RMSE. When calculating test RMSE, the predicted values will be found by predicting the response using the test data with the model fit using the train data. Test data should never be used to fit a model.
Set a seed of 22, and then split the Auto data into two datasets, one called auto_trn and one called auto_tst. The auto_trn data frame should contain 290 randomly chosen observations. The auto_tst data will contain the remaining observations. Hint: consider the following code:
set.seed(22)
auto_trn_idx = sample(1:nrow(Auto), 290)Fit a total of five models using the training data.
displacement as a predictor.For each model report the train and test RMSE. Arrange your results in a well-formatted markdown table. Argue that one of your models is the best for predicting the response.
For this exercise we will simulate data from the following model:
\[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5} + \epsilon_i \]
Where \(\epsilon_i \sim N(0, \sigma^2).\) Also, the parameters are known to be:
We will use samples of size n = 40.
We will verify the distribution of \(\hat{\beta}_1\) as well as investigate some hypothesis tests.
(a) We will first generate the \(X\) matrix and data frame that will be used throughout the exercise. Create the following nine variables:
x0: a vector of length n that contains all 1x1: a vector of length n that is randomly drawn from a normal distribution with a mean of 0 and a standard deviation of 2x2: a vector of length n that is randomly drawn from a uniform distribution between 0 and 4x3: a vector of length n that is randomly drawn from a normal distribution with a mean of 0 and a standard deviation of 1x4: a vector of length n that is randomly drawn from a uniform distribution between -2 and 2x5: a vector of length n that is randomly drawn from a normal distribution with a mean of 0 and a standard deviation of 2X: a matrix that contains x0, x1, x2, x3, x4, and x5 as its columnsC: the \(C\) matrix that is defined as \((X^\top X)^{-1}\)y: a vector of length n that contains all 0sim_data: a data frame that stores y and the five predictor variables. y is currently a placeholder that we will update during the simulation.Report the sum of the diagonal of C as well as the 5th row of sim_data. For this exercise we will use the seed 420. Generate the above variables in the order listed after running the code below to set a seed.
set.seed(400)
sample_size = 40(b) Create three vectors of length 2500 that will store results from the simulation in part (c). Call them beta_hat_1, beta_3_pval, and beta_5_pval.
(c) Simulate 2500 samples of size n = 40 from the model above. Each time update the y value of sim_data. Then use lm() to fit a multiple regression model. Each time store:
beta_hat_1beta_3_pvalbeta_5_pval(d) Based on the known values of \(X\), what is the true distribution of \(\hat{\beta}_1\)?
(e) Calculate the mean and variance of beta_hat_1. Are they close to what we would expect? Plot a histogram of beta_hat_1. Add a curve for the true distribution of \(\hat{\beta}_1\). Does the curve seem to match the histogram?
(f) What proportion of the p-values stored in beta_3_pval is less than 0.10? Is this what you would expect?
(g) What proportion of the p-values stored in beta_5_pval is less than 0.01? Is this what you would expect?