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Questions and Answers
Based on the regression output, which interpretation of the intercept is most accurate?
Based on the regression output, which interpretation of the intercept is most accurate?
- The intercept is always a meaningful value that can be directly interpreted.
- The intercept represents the predicted value of the response variable when all predictors are at their average values.
- The intercept should always be excluded from the regression model.
- The intercept represents the predicted value of the response variable when all predictors are zero. (correct)
In a multiple regression model predicting children's test scores, one of the predictors is whether the child's mother has a high school diploma (mom_hs). The coefficient for mom_hs:yes is 5.09. What is the correct interpretation of this coefficient?
In a multiple regression model predicting children's test scores, one of the predictors is whether the child's mother has a high school diploma (mom_hs). The coefficient for mom_hs:yes is 5.09. What is the correct interpretation of this coefficient?
- A child whose mother has a high school diploma is predicted to score 5.09 points higher on the test, regardless of other factors.
- A child whose mother has a high school diploma is predicted to score 5.09 points higher on the test, all else being equal. (correct)
- A child whose mother does not have a high school diploma is predicted to score 5.09 points higher on the test.
- A child whose mother has a high school diploma is predicted to score 5.09 points lower on the test, all else being equal.
In a multiple regression model, the coefficient for 'mother's IQ' (mom_iq) is 0.56. Which of the following is the correct interpretation of the slope for mom_iq?
In a multiple regression model, the coefficient for 'mother's IQ' (mom_iq) is 0.56. Which of the following is the correct interpretation of the slope for mom_iq?
- Kids whose mothers IQ's are one point higher tend to score on average 0.56 points higher. (correct)
- Kids whose mothers IQ's are one point lower tend to score on average 0.56 points higher.
- Kids whose mothers IQ's are zero tend to score on average 0.56 points higher.
- Kids whose mothers all have the same IQ score on average 0.56 points higher.
In a regression model, 'collinearity' refers to what condition?
In a regression model, 'collinearity' refers to what condition?
Why is adjusted R often preferred over R when comparing multiple regression models?
Why is adjusted R often preferred over R when comparing multiple regression models?
In a regression model, what is the correct formula for calculating R based on the sums of squares?
In a regression model, what is the correct formula for calculating R based on the sums of squares?
A regression model predicts poverty using the percentage of female-headed households. The R is 0.28. A second model is created that adds the percentage of the population that is white as a second predictor. The new R is 0.29, and the adjusted R is 0.26. What does this suggest?
A regression model predicts poverty using the percentage of female-headed households. The R is 0.28. A second model is created that adds the percentage of the population that is white as a second predictor. The new R is 0.29, and the adjusted R is 0.26. What does this suggest?
In multiple regression, how does the interpretation of a predictor variable's coefficient change compared to simple linear regression?
In multiple regression, how does the interpretation of a predictor variable's coefficient change compared to simple linear regression?
Which factor contributes to multicollinearity between explanatory variables?
Which factor contributes to multicollinearity between explanatory variables?
According to the data chart, which region is the reference level?
According to the data chart, which region is the reference level?
Based on data charts, which region has the lowest poverty percentage?
Based on data charts, which region has the lowest poverty percentage?
If weights of 80% of books can be predicted accurately, what are we using?
If weights of 80% of books can be predicted accurately, what are we using?
Based on scatterplot and weights, if Books are 10 $cm^3$ over average, what can we expect?
Based on scatterplot and weights, if Books are 10 $cm^3$ over average, what can we expect?
What does the relationship tell us between volume and weight?
What does the relationship tell us between volume and weight?
Which book is underestimated based on volume?
Which book is underestimated based on volume?
What is the volume of the hardcover book?
What is the volume of the hardcover book?
When determining reference level for books, what is the paperback book symbol?
When determining reference level for books, what is the paperback book symbol?
Based on the Coefficients and the data, what is true about estimating the cover when given the volume?
Based on the Coefficients and the data, what is true about estimating the cover when given the volume?
What roles should the variable cover type and the volume have in the regression model?
What roles should the variable cover type and the volume have in the regression model?
According to the slopes of volume, what do they express?
According to the slopes of volume, what do they express?
According to this data, if hardcover books have no volume what do we expect?
According to this data, if hardcover books have no volume what do we expect?
When predicting a weight what does this expression estimate? $\widehat{weight} = 197.96 + 0.72 volume 184.05 cover : pb$
When predicting a weight what does this expression estimate? $\widehat{weight} = 197.96 + 0.72 volume 184.05 cover : pb$
Based on all equals, kids who work score less with, or more to moms that do not work. What is the correct one?
Based on all equals, kids who work score less with, or more to moms that do not work. What is the correct one?
What are the benefits from 'collinearity' complication of estimation model in Observational data?
What are the benefits from 'collinearity' complication of estimation model in Observational data?
Which variable increases in model $R^2$?
Which variable increases in model $R^2$?
If a variable isn't unrelated, what increases in data?
If a variable isn't unrelated, what increases in data?
Why should we choose models with higher $R^2_{adj}$ over others?
Why should we choose models with higher $R^2_{adj}$ over others?
What statement is correct for sum of squares of $x$?
What statement is correct for sum of squares of $x$?
How do we calculate $R^2$?
How do we calculate $R^2$?
When calculating $R^2$, what should never be negative?
When calculating $R^2$, what should never be negative?
When looking at poverty percentages, which variable does not add information previously?
When looking at poverty percentages, which variable does not add information previously?
If using ANOVA function, what are we testing in variability?
If using ANOVA function, what are we testing in variability?
If we have three ways of calculating a single-predictor linear regression model, this would be considered what?
If we have three ways of calculating a single-predictor linear regression model, this would be considered what?
What are other measure for 'adj' and what does it stand for?
What are other measure for 'adj' and what does it stand for?
When is the intercept important to examine?
When is the intercept important to examine?
What will Adjusted R-squared penalize?
What will Adjusted R-squared penalize?
In a scenario where you are predicting a child's test score using multiple regression with mother's IQ (mom_iq), mother's work status (mom_work), and mother's education, how would you best describe the relationship between these variables?
In a scenario where you are predicting a child's test score using multiple regression with mother's IQ (mom_iq), mother's work status (mom_work), and mother's education, how would you best describe the relationship between these variables?
Suppose you are building a multiple regression model and find that two of your predictor variables, 'percentage of the population that is white' and 'percentage of female-headed households', are highly correlated. What is the primary concern with this situation?
Suppose you are building a multiple regression model and find that two of your predictor variables, 'percentage of the population that is white' and 'percentage of female-headed households', are highly correlated. What is the primary concern with this situation?
In a multiple regression model, after adding a new variable, you observe that the adjusted R-squared value decreases. What does this indicate about the added variable?
In a multiple regression model, after adding a new variable, you observe that the adjusted R-squared value decreases. What does this indicate about the added variable?
Why is calculating explained variability and total variability useful, especially when we can directly calculate $R^2$ as the square of the correlation coefficient in simple linear regression?
Why is calculating explained variability and total variability useful, especially when we can directly calculate $R^2$ as the square of the correlation coefficient in simple linear regression?
In a multiple regression model predicting book weight from volume and cover type (hardcover or paperback), what is the most accurate interpretation of the intercept?
In a multiple regression model predicting book weight from volume and cover type (hardcover or paperback), what is the most accurate interpretation of the intercept?
When interpreting the slope of a predictor variable in a multiple regression model, what crucial assumption must be made?
When interpreting the slope of a predictor variable in a multiple regression model, what crucial assumption must be made?
What is the main purpose of including a reference level when dealing with categorical variables (like region: Northeast, Midwest, South, West) in a regression model?
What is the main purpose of including a reference level when dealing with categorical variables (like region: Northeast, Midwest, South, West) in a regression model?
According to the formula $R_{adj}^2 = 1 - (\frac{SS_{Error}}{SS_{Total}} \times \frac{n-1}{n-p-1})$, what relationship is represented?
According to the formula $R_{adj}^2 = 1 - (\frac{SS_{Error}}{SS_{Total}} \times \frac{n-1}{n-p-1})$, what relationship is represented?
What is the purpose of a model with a single predictor, and having three ways to calculate?
What is the purpose of a model with a single predictor, and having three ways to calculate?
Variables such as female head of household and _ white_ may be valuable for poverty, but do they suggest anything of the collinearity?
Variables such as female head of household and _ white_ may be valuable for poverty, but do they suggest anything of the collinearity?
Flashcards
Multiple Regression
Multiple Regression
A statistical method using multiple variables to predict an outcome.
Explanatory Variable
Explanatory Variable
In regression, it's the variable used to predict the outcome.
Reference Level
Reference Level
The baseline or starting point to which other categories are compared
Intercept
Intercept
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Slope
Slope
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Hardcover
Hardcover
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Collinearity
Collinearity
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Parsimonious Model
Parsimonious Model
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Adjusted R-squared
Adjusted R-squared
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Study Notes
- Simple linear regression is bivariate, involving two variables, y and x.
- Multiple linear regression involves multiple variables represented as y, x1, x2, and so on.
Poverty vs. Region
- Region is the explanatory variable with the east as the reference level
- Intercept represents the estimated average poverty percentage in eastern states, at 11.17%.
- Plugging in 0 for the explanatory variable yields the intercept value.
- Slope indicates that the estimated average poverty percentage in western states is 0.38% higher than in eastern states.
- The estimated average poverty percentage in western states is 11.55% (11.17 + 0.38).
- Plugging in 1 for the explanatory variable yields the slope value.
- Northeast is the reference level for poverty vs. region (northeast, midwest, west, south).
- Northeast has the lowest poverty percentage.
Weights of Books
- Weights of 80% of books can be accurately predicted using the regression model.
- Books that are 10 cm³ over average are expected to weigh 7g over average.
- The correlation between weight and volume is R = 0.802 = 0.64.
- The model underestimates the weight of the book with the highest volume.
Modeling Book Weights with Volume
- The intercept is 107.67931.
- Volume is 0.70864.
- Residual standard error is 123.9, based on 13 degrees of freedom.
- Multiple R-squared is 0.8026, with an adjusted R-squared of 0.7875.
- F-statistic is 52.87 on 1 and 13 DF, and the p-value is 6.262e-06.
- Paperbacks generally weigh less than hardcover books after taking volume into account.
- In a model of book weights using volume and cover type, the intercept is 197.96284.
- Volume is 0.71795.
- Cover: PB(Paper Back) is -184.04727.
- The residual standard error is 78.2 on 12 degrees of freedom.
- Multiple R-squared is 0.9275, and adjusted R-squared is 0.9154.
- The F-statistic is 76.73 with 2 and 12 DF, and the p-value is 1.455e-07.
- Hardcover is the reference level, paperback book (pb:paperback) is noted.
- The roles of variables in this regression model are: response: weight, explanatory: volume, cover type.
Linear Model for Book Weight
- For hardcover books (cover = 0): weight = 197.96 + 0.72 * volume.
- For paperback books (cover = 1): weight = 13.91 + 0.72 * volume.
- Books with 1 more cubic centimeter in volume tend to weigh about 0.72 grams more, all else being constant.
- The model predicts that paperback books weigh 184 grams less than hardcover books, all else being constant.
- Hardcover books with no volume are expected on average to weigh about 198 grams, which doesn't necessarily make sense in context.
Linear Model prediction
- The correct calculation for predicting the weight of a paperback book that is 600 cm3 is 197.96 + 0.72 * 600 - 184.05 * 1.
Modeling Kid's Test Scores
- Kids with mothers whose IQs are one point higher tend to score on average 0.56 points higher, all else held constant.
- Kids whose moms have not attended high school, did three years of the kid's life, have an IQ of 0 expected on average to score 19.59, which doesn't necessarily make sense in context.
- Kids whose moms worked during the first three years of the kid's life are estimated to score 2.54 points higher than those whose moms did not work, keeping all else equal.
- Two predictor variables are collinear when they are correlated.
- Collinearity complicates model estimation.
- Predictors are also called explanatory or independent variables and ideally, they would be independent of each other.
R2
- The addition of a variable in a model will ensure R2 increases
- The third approach, uses the ratio of explained and unexplained variability
Adjusted R2
- The adjusted R2 equation is R2adj = 1 - (SSError / SSTotal) * (n-1 / n-p-1)
- n = number of cases
- p = number of predictors (explanatory variables) in the model
- R2adj will always be smaller than R2 because p is never negative
- Applying R2adj includes a penalty for the number of predictors included in the model.
- Models with higher R2adj over others are prefered.
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