class: center, middle, inverse, title-slide .title[ # EDUC 847 Winter 24 ] .subtitle[ ## Week 2 - Fundamentals of Linear Modeling ] .author[ ### Eric Brewe <br> Professor of Physics at Drexel University <br> ] .date[ ### 22 January 2024, last update: 2024-01-22 ] --- class: center, middle # Let's start where we ended last class...Scatter Plots Start by getting the data into R ```r #This loads the csv and saves it as a dataframe titled week_1_data week_1_data <- read_csv(here("static/slides/EDUC_847/data", "EDUC_847_Survey.csv")) ``` --- # Let's start cleaning up. ## First, we don't need most of that data There is a ton of data there that doesn't make sense for us to keep around. We will use the '%>%' (pipe) operator and the verb select ```r week_1_data %>% * select(ID,Q2:Q6) -> w1df glimpse(w1df) ``` ``` ## Rows: 10 ## Columns: 6 ## $ ID <dbl> 523, 876, 490, 166, 318, 626, 655, 182, 219, 305 ## $ Q2 <chr> "I am more of a", "{\"ImportId\":\"QID4\"}", NA, "1", "2", "1", "1"… ## $ Q3 <chr> "My favorite dessert is", "{\"ImportId\":\"QID2\"}", NA, "1", "1", … ## $ Q4 <chr> "Which of the following beverages have you had this week (select al… ## $ Q5 <chr> "How many pages were in the book that is physically closest to you?… ## $ Q6 <chr> "Estimate the weight of this cow. Please enter a number.", "{\"Impo… ``` --- # Let's keep cleaning up. Notice there are two rows of data that are just the questions and something from Qualtrics. Let's get rid of those. ```r w1df %>% filter(row_number() >3 ) -> w1df ``` --- # Let's lump all this together so we don't have to do it again. ```r week_1_data %>% select(ID,Q2:Q6) %>% filter(row_number() >3 ) %>% * mutate(across(c(Q2:Q3,Q5:Q6), as.numeric)) -> w1df ``` --- # Let's look at books vs. weight of cow ### Is there a relationship between length of book and estimates on weight of cow? .pull-left[ ### Making a scatter plot to explore ```r w1df %>% select(Q5:Q6) %>% ggplot(aes(x = Q5, y = Q6)) + geom_point() + theme_minimal() + labs(x = "Pages", y = "Weight of Cow", title = "Kick Ass Scatterplot!") ``` ] .pull-right[ <!-- --> ] A **residual** is the distance between the Y value for each measurement, and the predicted Y value. --- # Let's investigate the residuals for our data .pull-left[ ```r w1df %>% select(Q5,Q6) ``` ``` ## # A tibble: 7 × 2 ## Q5 Q6 ## <dbl> <dbl> ## 1 273 4389 ## 2 32 1200 ## 3 383 850 ## 4 352 1500 ## 5 869 1700 ## 6 56 1400 ## 7 81 1256 ``` ] .pull-right[ <!-- --> ] --- # Let's make a linear regression model ### We want the estimated weight of the cow as predicted by the length of the book closest to the survey respondent. ```r *Model1 <- lm(Q6 ~ Q5, data = w1df) Model1 ``` ``` ## ## Call: ## lm(formula = Q6 ~ Q5, data = w1df) ## ## Coefficients: ## (Intercept) Q5 ## 1668.7235 0.3001 ``` --- # Let's clean up our language... In this situation, Q5 (the Estimated weight of the cow) is the Outcome Variable and Q6 (the number of pages in the closest book) is the Predictor variable. Outcome Variable is often referred to as independent variable Predictor Variable is often referred to as the dependent variable. I hate this language. Predictor and Outcome are much more descriptive. --- # Let's figure out what this means ```r Model1 ``` ``` ## ## Call: ## lm(formula = Q6 ~ Q5, data = w1df) ## ## Coefficients: ## (Intercept) Q5 ## 1668.7235 0.3001 ``` **Intercept** = if the closest book had 0 pages in it, this would be the estimated weight of the cow that we predict. **Q6 coefficient** = If we increase the number of pages in the book, the value of the coeffient tells us how much we would increase the estimated weight of the cow by --- # Let's check out those residuals... .pull-left[ <!-- --> ] .pull-right[ ```r augment(Model1) %>% select(Q5:.resid) ``` ``` ## # A tibble: 7 × 3 ## Q5 .fitted .resid ## <dbl> <dbl> <dbl> ## 1 273 1751. 2638. ## 2 32 1678. -478. ## 3 383 1784. -934. ## 4 352 1774. -274. ## 5 869 1929. -229. ## 6 56 1686. -286. ## 7 81 1693. -437. ``` ] --- # Let's see how much better the model fits ```r augment(Model1) %>% select(Q5:.resid) %>% * mutate(SqResid = .resid^2) ``` ``` ## # A tibble: 7 × 4 ## Q5 .fitted .resid SqResid ## <dbl> <dbl> <dbl> <dbl> ## 1 273 1751. 2638. 6960935. ## 2 32 1678. -478. 228795. ## 3 383 1784. -934. 871700. ## 4 352 1774. -274. 75266. ## 5 869 1929. -229. 52662. ## 6 56 1686. -286. 81526. ## 7 81 1693. -437. 190994. ``` --- # Let's see how much better the model fits ```r augment(Model1) %>% select(Q5:.resid) %>% mutate(SqResid = .resid^2) %>% tally(SqResid) ``` ``` ## # A tibble: 1 × 1 ## n ## <dbl> ## 1 8461878. ``` --- # Let's take a look at R² **R²** is the measure of how well the model fits `\(R^{2} = 1 - \frac{\sum_{i} (Y_{Pred} - Y_{i})^2}{\sum_{i} (Y_{i}-\overline{Y})^2}\)` ```r augment(Model1) %>% select(Q6:.resid) %>% mutate(SqResid = .resid^2) %>% mutate(SqDiffMean = (Q6 - mean(Q6))^2) -> resid_df glimpse(resid_df) ``` ``` ## Rows: 7 ## Columns: 6 ## $ Q6 <dbl> 4389, 1200, 850, 1500, 1700, 1400, 1256 ## $ Q5 <dbl> 273, 32, 383, 352, 869, 56, 81 ## $ .fitted <dbl> 1750.642, 1678.326, 1783.649, 1774.347, 1929.481, 1685.527,… ## $ .resid <dbl> 2638.3584, -478.3256, -933.6489, -274.3468, -229.4810, -285… ## $ SqResid <dbl> 6960935.14, 228795.43, 871700.18, 75266.17, 52661.52, 81525… ## $ SqDiffMean <dbl> 6930432.327, 309612.755, 821612.755, 65755.612, 3184.184, 1… ``` --- # Let's take a look at R² `\(R^{2} = 1 - \frac{\sum_{i} (Y_{Pred} - Y_{i})^2}{\sum_{i} (Y_{i}-\overline{Y})^2}\)` .pull-left[ ```r SquaredResiduals = sum(resid_df$SqResid) SquaredResiduals ``` ``` ## [1] 8461878 ``` ```r SquaredDiffMean = sum(resid_df$SqDiffMean) SquaredDiffMean ``` ``` ## [1] 8508068 ``` ```r r2 = 1-(SquaredResiduals/SquaredDiffMean) r2 ``` ``` ## [1] 0.005428873 ``` --- # Let's compare to the R<sup>2<\sup> ```r SquaredResiduals ``` ``` ## [1] 8461878 ``` ```r SquaredDiffMean ``` ``` ## [1] 8508068 ``` ```r r2 ``` ``` ## [1] 0.005428873 ``` ```r summary(Model1) ``` ``` ## ## Call: ## lm(formula = Q6 ~ Q5, data = w1df) ## ## Residuals: ## 1 2 3 4 5 6 7 ## 2638.4 -478.3 -933.6 -274.3 -229.5 -285.5 -437.0 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 1668.7235 723.6088 2.306 0.0692 . ## Q5 0.3001 1.8163 0.165 0.8753 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 1301 on 5 degrees of freedom ## Multiple R-squared: 0.005429, Adjusted R-squared: -0.1935 ## F-statistic: 0.02729 on 1 and 5 DF, p-value: 0.8753 ``` --- # Let's recap We can... - Fit a regression model to two continuous variables - Calculate the residuals - Calculate the squared residuals - Use the residuals to calculate the r<sup>2<\sup>