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Chapter 12 Multiple Regression

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- Learn….
To use Multiple Regression Analysis to predict a response variable using more than one explanatory variable.

How Can We Use Several Variables to Predict a Response?

- The model that contains only two variables, x and y, is called a bivariate model

- The regression equation for the bivariate model is:

- Suppose there are two predictors, denoted by x1 and x2
- This is called a multiple regression model

- The regression equation for this multiple regression model with two predictors is:

- The multiple regression model relates the mean µy of a quantitative response variable y to a set of explanatory variables x1, x2,….

- Example: For three explanatory variables, the multiple regression equation is:

- Example: The sample prediction equation with three explanatory variables is:

- The data set “house selling prices” contains observations on 100 home sales in Florida in November 2003
- A multiple regression analysis was done with selling price as the response variable and with house size and lot size as the explanatory variables

- Output from the analysis:

- Prediction Equation:
where y = selling price, x1=house size and x2 = lot size

- One house listed in the data set had house size = 1240 square feet, lot size = 18,000 square feet and selling price = $145,000
- Find its predicted selling price:

- Find its residual:
- The residual tells us that the actual selling price was $37,724 higher than predicted

- You should not use many explanatory variables in a multiple regression model unless you have lots of data
- A rough guideline is that the sample size n should be at least 10 times the number of explanatory variables

- Always look at the data before doing a multiple regression
- Most software has the option of constructing scatterplots on a single graph for each pair of variables
- This is called a scatterplot matrix

- The simplest way to interpret a multiple regression equation looks at it in two dimensions as a function of a single explanatory variable
- We can look at it this way by fixing values for the other explanatory variable(s)

Example using the housing data:

- Suppose we fix x1 = house size at 2000 square feet
- The prediction equation becomes:

- Since the slope coefficient of x2 is 2.84, the predicted selling price for 2000 square foot houses increases by $2.84 for every square foot increase in lot size
- For a 1000 square-foot increase in lot size, the predicted selling price of 2000 sq. ft. houses increases by 1000(2.84) = $2840

Example using the housing data:

- Suppose we fix x2 = lot size at 30,000 square feet
- The prediction equation becomes:

- Since the slope coefficient of x1 is 53.8, the predicted selling price for houses with a lot size of 30,000 sq. ft. increases by $53.80 for every square foot increase in house size

- In summary, an increase of a square foot in house size has a larger impact on the selling price ($53.80) than an increase of a square foot in lot size ($2.84)
- We can compare slopes for these explanatory variables because their units of measurement are the same (square feet)
- Slopes cannot be compared when the units differ

- The multiple regression model assumes that the slope for a particular explanatory variable is identical for all fixed values of the other explanatory variables

- For example, the coefficient of x1 in the prediction equation:
is 53.8 regardless of whether we plug in x2 = 10,000 or x2 = 30,000 or x2 = 50,000

- In multiple regression, a slope describes the effect of an explanatory variable while controlling effects of the other explanatory variables in the model

- Bivariate regression has only a single explanatory variable
- A slope in bivariate regression describes the effect of that variable while ignoring all other possible explanatory variables

- One of the main uses of multiple regression is to identify potential lurking variables and control for them by including them as explanatory variables in the model

For all students at Walden Univ., the prediction equation for y = college GPA and x1= H.S. GPA and x2= study time is:

- Find the predicted college GPA of a student who has a H.S. GPA of 3.5 and who studies 3 hrs. per day.
- 3.67
- 3.005
- 3.175
- 3.4

For all students at Walden Univ., the prediction equation for y = college GPA and x1= H.S. GPA and x2= study time is:

- For students with fixed study time, what is the change in predicted college GPA when H.S. GPA increases from 3.0 to 4.0?
- 1.13
- 0.0078
- 0.643
- 1.00

Extending the Correlation and R-Squared for Multiple Regression

- To summarize how well a multiple regression model predicts y, we analyze how well the observed y values correlate with the predicted y values
- The multiple correlation is the correlation between the observed y values and the predicted y values
- It is denoted by R

- For each subject, the regression equation provides a predicted value
- Each subject has an observed y-value and apredicted y-value

- The correlation computed between all pairs of observed y-values and predicted y-values is the multiple correlation, R
- The larger the multiple correlation, the better are the predictions of y by the set of explanatory variables

- The R-value always falls between 0 and 1
- In this way, the multiple correlation ‘R’ differs from the bivariate correlation ‘r’ between y and a single variable x, which falls between -1 and +1

- For predicting y, the square of R describes the relative improvement from using the prediction equation instead of using the sample mean, y

- The error in using the prediction equation to predict y is summarized by the residual sum of squares:

- The error in using y to predict y is summarized by the total sum of squares:

- The proportional reduction in error is:

- The better the predictions are using the regression equation, the larger R2 is
- For multiple regression, R2 is the square of the multiple correlation, R

- For the 100 observations on y = selling price, x1 = house size, and x2 = lot size, a table, called the ANOVA (analysis of variance) table was created
- The table displays the sums of squares in the SS column

- The R2value can be created from the sums of squares in the table

- Using house size and lot size together to predict selling price reduces the prediction error by 71%, relative to using y alone to predict selling price

- Find and interpret the multiple correlation
- There is a strong association between the observed and the predicted selling prices
- House size and lot size very much help us to predict selling prices

- If we used a bivariate regression model to predict selling price with house size as the predictor, the r2 value would be 0.58
- If we used a bivariate regression model to predict selling price with lot size as the predictor, the r2 value would be 0.51

- The multiple regression model has R2 0.71, so it provides better predictions than either bivariate model

- The previous example showed that R2 for the multiple regression model was larger than r2 for a bivariate model using only one of the explanatory variables
- A key factor of R2 is that it cannot decrease when predictors are added to a model

- R2 falls between 0 and 1
- The larger the value, the better the explanatory variables collectively predict y
- R2 =1 only when all residuals are 0, that is, when all regression predictions are prefect
- R2 = 0 when the correlation between y and each explanatory variable equals 0

- R2 gets larger, or at worst stays the same, whenever an explanatory variable is added to the multiple regression model
- The value of R2 does not depend on the units of measurement

- The single predictor in the data set that is most strongly associated with y is the house’s real estate tax assessment
- (r2 = 0.679)

- When we add house size as a second predictor, R2 goes up from 0.679 to 0.730
- As other predictors are added, R2 continues to go up, but not by much

- R2 does not increase much after a few predictors are in the model
- When there are many explanatory variables but the correlations among them are strong, once you have included a few of them in the model, R2 usually doesn’t increase muchmore when you add additional ones

- This does not mean that the additional variables are uncorrelated with the response variable
- It merely means that they don’t add much new power for predicting y, given the values of the predictors already in the model

In a data set used to predict body weight (in pounds), three predictors were used: height, percent body fat and age.

Their correlations with total body weight were:

Height: 0.745 Percent Body fat: 0.390 Age: -0.187

- Which explanatory variable gives by itself the best prediction of weight?
- Height
- Percent body fat
- Age

In a data set used to predict body weight (in pounds), three predictors were used: height, percent body fat and age.

Their correlations with total body weight were:

Height: 0.745 Percent Body fat: 0.390 Age: -0.187

- With height as the sole predictor, what is r2?
- .745
- .555
- .625
- .825

In a data set used to predict body weight (in pounds), three predictors were used: height, percent body fat and age.

Their correlations with total body weight were:

Height: 0.745 Percent Body fat: 0.390 Age: -0.187

- If Percent Body Fat is added to the model R2 = 0.66. If Age is then added to the model R2=0.67. Once you know height and % body fat, does age seem to help in predicting weight?
- No
- Yes

How Can We Use Multiple Regression to Make Inferences?

- Assumptions required when using a multiple regression model to make inferences about the population:
- The regression equation truly holds for the population means
- This implies that there is a straight-line relationship between the mean of y and each explanatory variable, with the same slope at each value of the other predictors

- Assumptions required when using a multiple regression model to make inferences about the population:
- The data were gathered using randomization
- The response variable y has a normal distribution at each combination of values of the explanatory variables, with the same standard deviation

- Consider a particular parameter, β1
- If β1= 0, the mean of y is identical for all values of x1, at fixed values of the other explanatory variables
- So, H0: β1= 0 states that y and x1 are statistically independent, controlling for the other variables
- This means that once the other explanatory variables are in the model, it doesn’t help to have x1 in the model

- Assumptions:
- Each explanatory variable has a straight-line relation with µy with the same slope for all combinations of values of other predictors in the model
- Data gathered with randomization
- Normal distribution for y with same standard deviation at each combination of values of other predictors in model

- Hypotheses:
- H0: β1= 0
- Ha: β1≠ 0
- When H0 is true, y is independent of x1, controlling for the other predictors

- Test Statistic:

- P-value: Two-tail probability from t- distribution of values larger than observed t test statistic (in absolute value)
The t-distribution has:

df = n – number of parameters in the regression equation

- Conclusion: Interpret P-value; compare to significance level if decision needed

- The “College Athletes” data set comes from a study of 64 University of Georgia female athletes
- The study measured several physical characteristics, including total body weight in pounds (TBW), height in inches (HGT), the percent of body fat (%BF) and age

- The results of fitting a multiple regression model for predicting weight using the other variables:

- Interpret the effect of age on weight in the multiple regression equation:

- The slope coefficient of age is -0.96
- For athletes having fixed values for x1 and x2,the predicted weight decreases by 0.96 pounds for a 1-year increase in age, and the ages vary only between 17 and 23

- Run a hypothesis test to determine whether age helps to predict weight, if you already know height and percent body fat

- Assumptions:
- The 64 female athletes were a convenience sample, not a random sample
- Caution should be taken when making inferences about all female college athletes

- Hypotheses:
- H0: β3= 0
- Ha: β3≠ 0

- Test statistic:

- P-value: This value is reported in the output as 0.14
- Conclusion:
- The P-value of 0.14 does not give much evidence against the null hypothesis that β3 = 0
- Age does not significantly predict weight if we already know height and % body fat

- A 95% confidence interval for a β slope parameter in multiple regression equals:
- The t-score has:
df = (n - # of parameters in the model)

- Construct and interpret a 95% CI for β3, the effect of age while controlling for height and % body fat

- At fixed values of x1 and x2, we infer that the population mean of weight changes very little (and maybe not at all) for a 1-year increase in age
- The confidence interval contains 0
- Age may have no effect on weight, once we control for height and % body fat

- A standard deviation parameter, σ, describes variability of the observations around the regression equation
- Its sample estimate is:

- Anova Table for the “college athletes” data set:

- For female athletes at particular values of height, % of body fat, and age, estimate the standard deviation of their weights
- Begin by finding the Mean Square Error:
- Notice that this value (102.2) appears in the MS column in the ANOVA table

- The standard deviation is:
- This value is also displayed in the ANOVA table
- For athletes with certain fixed values of height, % body fat, and age, the weights vary with a standard deviation of about 10 pounds

- If the conditional distributions of weight are approximately bell-shaped, about 95% of the weight values fall within about 2s = 20 pounds of the true regression line

- Example: With 3 predictors in a model, we can check this by testing:

- The test statistic for H0 is denoted by F

- When H0 is true, the expected value of the F test statistic is approximately 1
- When H0 is false, F tends to be larger than 1
- The larger the F test statistic, the stronger the evidence against H0

- Assumptions: Multiple regression equation holds, data gathered randomly, normal distribution for y with same standard deviation at each combination of predictors

- Test statistic:

- P-value: Right-tail probability above observed F-test statistic value from F- distribution with:
- df1 = number of explanatory variables
- df2 = n – (number of parameters in regression equation)

- Conclusion: The smaller the P-value, the stronger the evidence that at least one explanatory variable has an effect on y
- If a decision is needed, reject H0 if P-value ≤ significance level, such as 0.05

- For the 64 female college athletes, the regression model for predicting y = weight using x1 = height, x2 = % body fat and x3 = age is summarized in the ANOVA table on the next page

- Use the output in the ANOVA table to test the hypothesis:

- The observed F statistic is 40.48
- The corresponding P-value is 0.000
- We can reject H0at the 0.05 significance level
- We conclude that at least one predictor has an effect on weight

- The F-test tells us that at least one explanatory variable has an effect
- If the explanatory variables are chosen sensibly, at least one should have some predictive power
- The F-test result tells us whether there is sufficient evidence to make it worthwhile to consider the individual effects, using t-tests

- The individual t-tests identify which of the variables are significant (controlling for the other variables)

- If a variable turns out not to be significant, it can be removed from the model
- In this example, ‘age’ can be removed from the model

Checking a Regression Model Using Residual Plots

- The regression equation approximates well the true relationship between the predictors and the mean of y
- The data were gathered randomly
- y has a normal distribution with the same standard deviation at each combination of predictors

- To test Assumption 3 (the conditional distribution of y is normal at any fixed values of the explanatory variables):
- Construction a histogram of the standardized residuals
- The histogram should be approximately bell-shaped
- Nearly all the standardized residuals should fall between -3 and +3. Any residual outside these limits is a potential outlier

- For the house selling price data, a MINITAB histogram of the standardized residuals for the multiple regression model predicting selling price by the house size and the lot size was created and is displayed on the following page

- The residuals are roughly bell shaped about 0
- They fall between about -3 and +3
- No severe nonnormality is indicated

- Plots of residuals against each explanatory variable help us check for potential problems with the regression model
- Ideally, the residuals should fluctuate randomly about 0
- There should be no obvious change in trend or change in variation as the values of the explanatory variable increases

How Can Regression Include Categorical Predictors?

- Regression models specify categories of a categorical explanatory variable using artificial variables, called indicator variables
- The indicator variable for a particular category is binary
- It equals 1 if the observation falls into that category and it equals 0 otherwise

- In the house selling prices data set, the city region in which a house is located is a categorical variable
- The indicator variable x for region is
- x = 1 if house is in NW (northwest region)
- x = 0 if house is not in NW

- The coefficient β of the indicator variable x is the difference between the mean selling prices for homes in the NW and for homes not in the NW

- Output from the regression model for selling price of home using house size and region

- Find and plot the lines showing how predicted selling price varies as a function of house size, for homes in the NW and for homes no in the NW

- The regression equation from the MINITAB output is:

- For homes not in the NW, x2 = 0
- The prediction equation then simplifies to:

- For homes in the NW, x2 = 1
- The prediction equation then simplifies to:

- Both lines have the same slope, 78
- For homes in the NW and for homes not in the NW, the predicted selling price increases by $78 for each square-foot increase in house size
- The figure portrays a separate line for each category of region (NW, not NW)

- The coefficient of the indicator variable is 30569
- For any fixed value of house size, we predict that the selling price is $30,569 higher for homes in the NW

- The line for homes in the NW is above the line for homes not in the NW
- The predicted selling price is higher for homes in the NW
- The P-value of 0.000 for the test for the coefficient of the indicator variable suggests that this difference is statistically significant

- For two explanatory variables, interaction exists between them in their effects on the response variable when the slope of the relationship between µy and one of them changes as the value of the other changes

How Can We Model a Categorical Response?

- When y is categorical, a different regression model applies, called a logistic regression

- A voter’s choice in an election (Democrat or Republican), with explanatory variables: annual income, political ideology, religious affiliation, and race
- Whether a credit card holder pays their bill on time (yes or no), with explanatory variables: family income and the number of months in the past year that the customer paid the bill on time

- Denote the possible outcomes for y as 0 and 1
- Use the generic terms failure (for outcome = 0) and success (for outcome =1)
- The population mean of the scores equals the population proportion of ‘1’ outcomes (successes)
- That is, µy = p

- The proportion, p, also represents the probability that a randomly selected subject has a success outcome

- The straight-line model is usually inadequate
- A more realistic model has a curved S-shape instead of a straight-line trend

- A regression equation for an S-shaped curve for the probability of success p is:

- An Italian study with 100 randomly selected Italian adults considered factors that are associated with whether a person possesses at least one travel credit card
- The table on the next page shows results for the first 15 people on this response variable and on the person’s annual income (in thousands of euros)

- Let x = annual income and let y = whether the person possesses a travel credit card (1 = yes, 0 = no)

- Substituting the α and β estimates into the logistic regression model formula yields:

- Find the estimated probability of possessing a travel credit card at the lowest and highest annual income levels in the sample, which were x = 12 and x = 65

- For x = 12 thousand euros, the estimated probability of possessing a travel credit card is:

- For x = 65 thousand euros, the estimated probability of possessing a travel credit card is:

- Annual income has a strong positive effect on having a credit card
- The estimated probability of having a travel credit card changes from 0.09 to 0.97 as annual income changes over its range

- A three-variable contingency table from a survey of senior high-school students in shown on the next page
- The students were asked whether they had ever used: alcohol, cigarettes or marijuana

- Let y indicate marijuana use, coded: (1 = yes, 0 = no)
- Let x1 be an indicator variable for alcohol use (1 = yes, 0 = no)
- Let x2 be an indicator variable for cigarette use (1 = yes, 0 = no)

- The logistic regression prediction equation is:

- For those who have not used alcohol or cigarettes, x1= x2 = 0 and:

- For those who have used alcohol and cigarettes, x1= x2 = 1 and:

- The probability that students have tried marijuana seems to depend greatly on whether they’ve used alcohol and cigarettes