library(learnr) library(gradethis) library(dplyr) library(stringr) library(ggplot2) library(moderndive) library(ggcheck) knitr::opts_chunk$set(echo = FALSE) gradethis_setup( pass.praise = TRUE, fail.hint = FALSE, fail_code_feedback = FALSE, fail.encourage = TRUE, maybe_code_feedback = FALSE) cars2008 <- dplyr::filter(mpg, year==2008) wine_data <- DataScienceExercises::wine2dine
Content quiz I
quiz( question(paste( "Can it be that adding a new variable to a regression", "equation changes the estimates for the other variable?"), answer( paste0("Yes, this is true for all practically relevant cases."), correct = TRUE), answer("No, adding a new variable does not change estimates for the other variables."), answer("It depends on the particular estimation method."), answer("It depends on the number of observations."), answer("It depends on the total number of variables in the regression equation."), allow_retry = TRUE, random_answer_order = TRUE), question( paste("What is a categorial variable?"), answer( paste0("A variable that can take one out of a pre-specified number of different values."), correct = TRUE), answer("A variable that is stored as `character`."), answer("A variable that can be interpreted as representing an economic category."), answer("A variable that cannot be considered within the context of linear regression analysis."), allow_retry = TRUE, random_answer_order = TRUE), question( paste("What does 'ceteris paribus' mean?"), answer( paste0("All other things held constant."), correct = TRUE), answer("F*** statistics!"), answer("All variables taken into account."), answer("Including the same effect of other variables."), answer("Not considering other effects."), allow_retry = TRUE, random_answer_order = TRUE), question( paste("What is the 'baseline level' in a regression with a categorial variable?"), answer( paste0("The level against which the effects of the other levels are reported."), correct = TRUE), answer("The most important result of the regression."), answer("The level that is reported as the first category in the regression."), answer("The most frequently occuring level of the variable in the data."), answer("The level with the most intuitive interpretation."), allow_retry = TRUE, random_answer_order = TRUE) )
Content quiz II
Consider the following regression output, in which kind is a
categorial variable that can take the values 'red' or 'white' and
alcohol provides information about the alcohol content of the whine.
lm(alcohol~kind, data = wine_data)
quiz( question( paste("What does the estimate for the intercept tells us?"), answer( paste0("The average alcohol content of red wine is 10.4%."), correct = TRUE), answer( paste0("The average alcohol content of white wine is 10.4%.")), answer( paste0("The average alcohol content of red wine is 10.4 percent ", "points below that of white whine")), answer( paste0("On average, changing the type of wine from white to red ", "is associated with an increase in the alcohol content of 10.4%")), allow_retry = TRUE, random_answer_order = TRUE), question( paste("How can we interpret the estimate `kindwhite`?"), answer( paste0("The average alcohol content of white wine is 0.09128 percent ", "points higher than that of red wine."), correct = TRUE), answer( paste0("The average alcohol content of white wine is 9.128%.")), answer( paste0("On average, changing the type of wine from red to white ", "is associated with an decrease in the alcohol content of 0.09128%" )), allow_retry = TRUE, random_answer_order = TRUE) )
Content quiz III
quiz( question( paste("Under which circumstances should you consider estimating", "an interaction model?"), answer( paste0("When you have more than one explanatory variable and you suspect ", "that the effect on one variable depends on the value of another."), correct = TRUE), answer("Never, its only useful for paedagogical purposes."), answer(paste0( "When you want to estimate the effect of variables that represent ", "some kind of interaction relationships between individuals." )), answer(paste0( "When some of the explanatory variables correlate strongly with ", "each other.")), allow_retry = TRUE, random_answer_order = TRUE), question( paste("When should you use a parallel slopes model instead of an interaction model?"), answer( paste0("When you have strong reason to believe that the effect of the ", "explanatory variable is the same across groups."), correct = TRUE), answer("Never, its only useful for paedagogical purposes."), answer("When there is no other specification without interaction effects available."), answer("It is generally the better choice because it is more efficient."), answer("When the sample size exceeds the parameters to be estimtated by factor 10 or more."), allow_retry = TRUE, random_answer_order = TRUE), question(paste( "What is the main advantage of \\bar{R}^2 over R^2?"), answer(paste0( "It decreases if you add an additional explanatory variable ", "that does not add to the explanatory power of the model."), correct = TRUE), answer(paste0( "You can still interpret it as the share of explained variation in the ", "dependent variable, even in the multiple regression context." )), answer(paste0( "It is more precise once more than one independent variable is considered." )), answer(paste0( "It takes into account the explanatory power of interaction effects ", "among explanatory variables." )), answer(paste0( "It is easier to compute." )), allow_retry = TRUE, random_answer_order = TRUE), question(paste("What is the main drawback of $\\bar{R}^2$ as compared to $R^2$?"), answer(paste0( "It increases if you add an additional explanatory variable ", "that does not add to the explanatory power of the model.")), answer(paste0( "You cannot interpret it as the share of explained variation in total ", "variation of the dependent variable." ), correct = TRUE), answer(paste0( "It becomes less precise the more explanatory variables are considered." )), answer(paste0( "It cannot take into account the explanatory power of interaction ", "effects among explanatory variables." )), answer("It requires more observations to be properly computed."), allow_retry = TRUE, random_answer_order = TRUE) )
Coding exercises I
Multiple regression with continuous variables
Consider the data set called cars2008, which contains information
about the fuel economy of various popular car models.
Now analyse how engine displacement and relate to highway miles per gallon for the different cars. To this end, conduct a linear regression with highway miles per gallon as the dependent, and engine displacement and number of cylinders as the independent variables.
Note: cars2008 is a subset of the data set mpg, which is built-in into R.
So if you need any information about the data (e.g. variable names), just
type ?mpg into the console.
Hint: To evaluate your result you need to return the regression object you have created.
cars2008 <- mpg %>% dplyr::select(all_of(c("hwy", "cyl", "displ", "manufacturer")))
lm(____)
lm(formula = ____, data = ____)
lm(formula = ____, data = cars2008)
lm(formula = ____~____+____, data = cars2008)
lm(formula = hwy~____+____, data = cars2008)
lm(formula = hwy~displ+____, data = cars2008)
lm(formula = hwy~displ+cyl, data = cars2008)
grade_this({ # Test whether the type of returned object is correct if(!inherits(.result, "lm")){ fail(message = paste( "Your code should be produce a linear model.", "Check out the function `lm()`!")) } correct_model <- lm(formula = hwy~displ+cyl, data = cars2008) call_made <- str_remove_all(as.character(.result$call), " ") # Check whether the correct data set has been used if (!"cars2008" %in% call_made){ fail("Your call should refer to the data set 'cars2008'!") } # Check whether the correct number of coefficients is returned if (length(coefficients(.result)) > 3){ fail("You used too many explanatory variables!") } if (length(coefficients(.result)) < 3){ fail("You used too few explanatory variables!") } # Check whether the regression formula was correct form_test_1 <- ("hwy~displ+cyl" %in% call_made) form_test_2 <- ("hwy~cyl+displ" %in% call_made) coefficient_test <- all.equal( sort(unname(coefficients(correct_model))), sort(unname(coefficients(.result)))) if ( (form_test_1 | form_test_2) & coefficient_test){ pass() } else{ fail("You did not specify the call of `lm()` correctly!") } })
Simple regression with a categorical variable
Consider again the data set called cars2008, but now analyze how
driving economy measured by highway miles per gallon differs across car
manufacturer.
To this end, implement a simple regression with highway miles per gallon as the dependent, and the manufacturer as the independent variable.
Note: cars2008 is a subset of the data set mpg, which is built-in into R.
So if you need any information about the data (e.g. variable names), just
type ?mpg into the console.
Hint: To evaluate your result you need to return the regression object you have created.
cars2008 <- mpg %>% dplyr::select(all_of(c("hwy", "cyl", "displ", "manufacturer")))
lm(____)
lm(formula = ____, data = ____)
lm(formula = ____, data = cars2008)
lm(formula = ____~____, data = cars2008)
lm(formula = hwy~____, data = cars2008)
lm(formula = hwy~manufacturer, data = cars2008)
grade_this({ # Test whether the type of returned object is correct if(!inherits(.result, "lm")){ fail(message = paste( "Your code should be produce a linear model.", "Check out the function `lm()`!")) } correct_model <- lm(formula = hwy~manufacturer, data = cars2008) call_made <- str_remove_all(as.character(.result$call), " ") # Check whether the correct data set has been used if (!"cars2008" %in% call_made){ fail("Your call should refer to the data set 'cars2008'!") } # Check whether the regression formula was correct form_test_1 <- ("hwy~manufacturer" %in% call_made) if (form_test_1){ pass() } else{ fail("You did not specify the call of `lm()` correctly!") } })
Now interpret the results to answer the following questions.
Hint: To get all the different values a variable takes you can use the
code unique(NAME_OF_DATA$NAME_OF_VARIABLE), e.g. unique(cars2008$hwy).
quiz( question( "The baseline manufacturer in this regression was...", answer("Audi", correct = TRUE), answer("Honda"), answer("Volkswagen"), answer("Activision"), allow_retry = TRUE, random_answer_order = TRUE), question( "The average highway miles per gallon of a Volkswagen are...", answer(paste("more than those achieved by the cars", "from the baseline manufacturer."), correct = TRUE), answer(paste("less than those achieved by the cars", "from the baseline manufacturer.")), answer(paste("the same as those achieved by the cars", "from the baseline manufacturer.")), allow_retry = TRUE, random_answer_order = TRUE), question( "Cars from Audi require smaller amounts of fuel than those from Chevrolet!", answer(paste("True!"), correct = TRUE), answer(paste("False!")), answer(paste("Impossible to say from this regression!")), allow_retry = TRUE, random_answer_order = TRUE) )
Multiple regression with both categorical and continuous variables
gdp_lifexp <- DataScienceExercises::gdplifexp2007 %>% dplyr::select(all_of(c("lifeExp", "continent", "gdpPercap"))) %>% dplyr::filter(continent != "Oceania") %>% dplyr::mutate(gdpPercap = log(gdpPercap/1000), lifeExp = log(lifeExp))
Consider the data set gdp_lifexp. It contains the following variables:
country: the countrylifeExp: average life expectancy in years in this country (logarithm)continent: the continent to which the country belongsgdpPercap: average GDP per capita in this country (1000 USD, logarithm)
Implement the following regression in R:
$$lifeExp = \beta_0 + \beta_1 continent + \beta_2 gdpPercap + \epsilon$$
Note: To evaluate your solution, return the result by simply calling
the object returned by lm
(i.e. without using moderndive::get_regression_table()).
gdp_lifexp_lm <- lm(____) gdp_lifexp_lm
gdp_lifexp_lm <- lm( formula = ____, data = ____) gdp_lifexp_lm
gdp_lifexp_lm <- lm( formula = ____, data = gdp_lifexp) gdp_lifexp_lm
gdp_lifexp_lm <- lm( formula = lifeExp ~ ____, data = gdp_lifexp) gdp_lifexp_lm
gdp_lifexp_lm <- lm( formula = lifeExp ~ continent + ____, data = gdp_lifexp) gdp_lifexp_lm
gdp_lifexp_lm <- lm( formula = lifeExp ~ continent + gdpPercap, data = gdp_lifexp) gdp_lifexp_lm
grade_this({ # Test whether the type of returned object is correct if(!inherits(.result, "lm")){ fail(message = paste( "Your code should be produce a linear model.", "Check out the function `lm()`!")) } correct_model <- lm( formula = lifeExp ~ continent + gdpPercap, data = gdp_lifexp) call_made <- str_remove_all(as.character(.result$call), " ") # Check whether the correct data set has been used if (!"gdp_lifexp" %in% call_made){ fail("Your call should refer to the data set 'gdp_lifexp'!") } # Check whether the correct number of coefficients is returned if (length(coefficients(.result)) > 5){ fail("You used too many explanatory variables!") } if (length(coefficients(.result)) < 5){ fail("You used too few explanatory variables!") } # Check whether the regression formula was correct form_test_1 <- ("lifeExp~continent+gdpPercap" %in% call_made) form_test_2 <- ("lifeExp~gdpPercap+continent" %in% call_made) coefficient_test <- all.equal( sort(unname(coefficients(correct_model))), sort(unname(coefficients(.result)))) if ( (form_test_1 | form_test_2) & coefficient_test){ pass() } else{ fail("You did not specify the call of `lm()` correctly!") } })
Coding exercises II
The parallel slopes model
teaching_evals <- moderndive::evals %>% select(all_of(c("age", "score", "gender")))
The model you implemented in the previous exercise is one example for what is
called a parallel slopes model.
To see why, first estimate the following model for the data set
teaching_evals, which contains the following variables:
score: the teaching score that the professor received in a teaching evaluation, ranging from1(very unsatisfactory) to5(excellent)age: the age of the professorgender: the gender of the professor (maleorfemale)
Estimate this model and return the regression object:
$$score = \beta_0 + \beta_1 age + \beta_2 gender + \epsilon$$
teaching_evals_psm <- lm(____) teaching_evals_psm
teaching_evals_psm <- lm( formula = ____, data = ____) teaching_evals_psm
teaching_evals_psm <- lm( formula = ____, data = teaching_evals) teaching_evals_psm
teaching_evals_psm <- lm( formula = score ~ ____, data = teaching_evals) teaching_evals_psm
teaching_evals_psm <- lm( formula = score ~ age + ____, data = teaching_evals) teaching_evals_psm
teaching_evals_psm <- lm( formula = score ~ age + gender, data = teaching_evals) teaching_evals_psm
grade_this({ # Test whether the type of returned object is correct if(!inherits(.result, "lm")){ fail(message = paste( "Your code should be produce a linear model.", "Check out the function `lm()`!")) } correct_model <- lm( formula = score ~ age + gender, data = teaching_evals) call_made <- str_remove_all(as.character(.result$call), " ") # Check whether the correct data set has been used if (!"teaching_evals" %in% call_made){ fail("Your call should refer to the data set 'teaching_evals'!") } # Check whether the correct number of coefficients is returned if (length(coefficients(.result)) > 3){ fail("You used too many explanatory variables!") } if (length(coefficients(.result)) < 3){ fail("You used too few explanatory variables!") } # Check whether the regression formula was correct form_test_1 <- ("score~age+gender" %in% call_made) form_test_2 <- ("score~gender+age" %in% call_made) coefficient_test <- all.equal( sort(unname(coefficients(correct_model))), sort(unname(coefficients(.result)))) if ( (form_test_1 | form_test_2) & coefficient_test){ pass() } else{ fail("You did not specify the call of `lm()` correctly!") } })
quiz( question( "Which of the following conclusions can you draw from the results?", answer("Older teachers tend to receive lower scores", correct = TRUE), answer("Male teachers tend to receive lower scores"), answer("Female teachers tend to receive lower scores", correct = TRUE), answer("Older teachers tend to receive lower scores when they are male"), answer("Older teachers tend to receive lower scores but only if they are female"), answer(paste("Both male and female teachers tend to receive lower", "scores when they are older, but the effect is much", "stronger for female teachers.")), answer(paste("Both male and female teachers tend to receive lower", "scores when they are older, but the effect is much", "stronger for male teachers.")), allow_retry = TRUE, random_answer_order = TRUE) )
To see why these models are called parallel slopes models, visualize the
model using ggplot2. To this end, create a scatter plot, with age being on
the x-axis, and score being on the y-axis.
Add a separate
regression line using the geom function geom_parallel_slopes(se = FALSE),
which is provided by the package moderndive.
To get separate lines for males and females, map the variable gender on the
aesthetic color.
teaching_evals <- moderndive::evals %>% select(all_of(c("age", "score", "gender")))
ggplot(____) + ____ + ____ + theme_bw()
ggplot( data = ____, mapping = ____ ) + ____ + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = ____ ) + ____ + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = ____, y = ____, color = ____) ) + ____ + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = ____, y = ____, color = ____) ) + geom_point() + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = ____) ) + geom_point() + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = gender) ) + geom_point() + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = gender) ) + geom_point() + geom_parallel_slopes(se = FALSE) + theme_bw()
grade_this({ fail_if(!ggcheck::is_ggplot(.result), message = paste0( "You did not define a `ggplot` object. Use the function ", "`ggplot2::ggplot()`. ", "Did you maybe forget to call and render the plot?" ) ) fail_if(!ggcheck::uses_data(.result, teaching_evals), message = paste0( "You should use the data set `teaching_evals`. You can pass ", "it to the argument `data` of `ggplot2::ggplot()`." ) ) fail_if(!uses_mappings(.result, aes(x = age, y = score, color = gender), exact = TRUE), message = paste0( "You should map the variable `age` to the x-axis, the variable ", "`score` to the y-axis, and the variabe `gender` to the color!" , "Use the argument `mapping` ", "of the `ggplot2::ggplot()` function, together with `ggplot2::aes()`." ) ) fail_if(!ggcheck::uses_geoms(.result, c("point"), exact = FALSE), message = paste0( "You were asked to construct a scatter plot with a regression ", "line, but are missing the points...what geom is missing?" ) ) if(!str_detect(str_replace_all(.user_code, "\\s", ""), "geom_parallel_slopes\\(se=FALSE\\)")){ fail("You are missing the call to 'geom_parallel_slopes(se = FALSE)'!") } pass() })
The interaction model
Now extend the model you estimated in the previous exercise to also account for
interaction effects between the variables age and gender.
teaching_evals <- moderndive::evals %>% select(all_of(c("age", "score", "gender")))
teaching_evals_int <- lm(____) teaching_evals_int
teaching_evals_int <- lm( formula = ____, data = ____) teaching_evals_int
teaching_evals_int <- lm( formula = ____, data = teaching_evals) teaching_evals_int
teaching_evals_int <- lm( formula = score ~ ____, data = teaching_evals) teaching_evals_int
teaching_evals_int <- lm( formula = score ~ age * ____, data = teaching_evals) teaching_evals_int
teaching_evals_int <- lm( formula = score ~ age * gender, data = teaching_evals) teaching_evals_int
grade_this({ # Test whether the type of returned object is correct if(!inherits(.result, "lm")){ fail(message = paste( "Your code should be produce a linear model.", "Check out the function `lm()`!")) } correct_model <- lm( formula = score ~ age * gender, data = teaching_evals) call_made <- str_remove_all(as.character(.result$call), " ") # Check whether the correct data set has been used if (!"teaching_evals" %in% call_made){ fail("Your call should refer to the data set 'teaching_evals'!") } # Check whether an interaction model was used if (!TRUE %in% str_detect(call_made, "\\*")){ fail(paste( "Remember that in interaction models you connect explanatory", "variables via `*` instead of `+`!")) } # Check whether the correct number of coefficients is returned if (length(coefficients(.result)) > 4){ fail("You used too many explanatory variables!") } if (length(coefficients(.result)) < 4){ fail("You used too few explanatory variables!") } # Check whether the regression formula was correct form_test_1 <- ("score~age*gender" %in% call_made) form_test_2 <- ("score~gender*age" %in% call_made) coefficient_test <- all.equal( sort(unname(coefficients(correct_model))), sort(unname(coefficients(.result)))) if ( (form_test_1 | form_test_2) & coefficient_test){ pass() } else{ fail(paste( "You did not specify the call of `lm()` correctly!")) } })
quiz( question( "Which of the following conclusions can you draw from the results?", answer("Male teachers tend to receive lower scores than female teachers"), answer("Female teachers tend to receive lower scores than male teachers"), answer("Older teachers tend to receive lower scores only when they are male"), answer("Older teachers tend to receive lower scores but only if they are female"), answer(paste("Both male and female teachers tend to receive lower", "scores when they are older, but the effect is much", "stronger for female teachers."), correct = TRUE), answer(paste("Both male and female teachers tend to receive lower", "scores when they are older, but the effect is much", "stronger for male teachers.")), allow_retry = TRUE, random_answer_order = TRUE) )
To see how this model differs from the parallel slopes model above, visualize the
model using ggplot2. To this end, create a scatter plot, with age being on
the x-axis, and score being on the y-axis.
Add a separate regression line using the geom function
geom_smooth(method = 'lm', se = FALSE)
To get separate lines for males and females, map the variable gender on the
aesthetic color.
teaching_evals <- moderndive::evals %>% select(all_of(c("age", "score", "gender")))
ggplot(____) + ____ + ____ + theme_bw()
ggplot( data = ____, mapping = ____ ) + ____ + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = ____ ) + ____ + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = ____, y = ____, color = ____) ) + ____ + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = ____, y = ____, color = ____) ) + geom_point() + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = ____) ) + geom_point() + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = gender) ) + geom_point() + ____ + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = gender) ) + geom_point() + geom_smooth(____) + theme_bw()
ggplot( data = teaching_evals, mapping = aes(x = age, y = score, color = gender) ) + geom_point() + geom_smooth(se=FALSE, method="lm") + theme_bw()
grade_this({ fail_if(!ggcheck::is_ggplot(.result), message = paste0( "You did not define a `ggplot` object. Use the function ", "`ggplot2::ggplot()`. ", "Did you maybe forget to call and render the plot?" ) ) fail_if(!ggcheck::uses_data(.result, teaching_evals), message = paste0( "You should use the data set `teaching_evals`. You can pass ", "it to the argument `data` of `ggplot2::ggplot()`." ) ) fail_if(!uses_mappings(.result, aes(x = age, y = score, color = gender), exact = TRUE), message = paste0( "You should map the variable `age` to the x-axis, the variable ", "`score` to the y-axis, and the variabe `gender` to the color!" , "Use the argument `mapping` ", "of the `ggplot2::ggplot()` function, together with `ggplot2::aes()`." ) ) fail_if(!ggcheck::uses_geoms(.result, c("point"), exact = FALSE), message = paste0( "You were asked to construct a scatter plot with a regression ", "line, but are missing the points...what geom is missing?" ) ) fail_if(!ggcheck::uses_geoms(.result, c("smooth"), exact = FALSE), message = paste0( "You were asked to construct a scatter plot with a regression ", "line, but are missing the regression line...what geom is missing?" ) ) if(!uses_geom_params(.result, "smooth", list(method = "lm"))){ fail(paste0( "You forgot the set the optional argument `method` for the smooth ", "geom to `'lm'`. Google can help you to find out what to do!")) } if(!uses_geom_params(.result, "smooth", list(se = FALSE))){ fail(paste0( "You forgot the set the optional argument `se` for the smooth ", "geom to `FALSE`.")) } pass() })
quiz( question( "Considering the case above, which of the two models is to be preferred?", answer("The parallel slopes model"), answer("The interaction model", correct = TRUE), answer("It does not matter"), allow_retry = TRUE, random_answer_order = TRUE) )
