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‎10-inference-for-regression.Rmd

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@@ -352,7 +352,7 @@ $$b_1 = \frac{\sum_{i=1}^n(x_i - \bar x)(y_i - \bar y)}{\sum_{i=1}^n(x_i - \bar
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Furthermore, an *estimator* for the standard deviation of $\epsilon_i$ is given by
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$s = \sqrt{\frac{\sum_{i=1}^n \left[y_i - (b_0 + b_1 \cdot x_i)\right]^2}{n-2}} = \sqrt{\frac{\sum_{i=1}^n \left(y_i - \widehat{y}_i\right)^2}{n-2}}.$
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$$s = \sqrt{\frac{\sum_{i=1}^n \left[y_i - (b_0 + b_1 \cdot x_i)\right]^2}{n-2}} = \sqrt{\frac{\sum_{i=1}^n \left(y_i - \widehat{y}_i\right)^2}{n-2}}.$$
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These or equivalent calculations are done in R when using the `lm()` function.
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For `old_faithful_2024` we get:
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```{r eval=FALSE}
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spotify_for_anova |>
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slice_sample(n = 10)
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slice_sample(n = 5)
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```
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(ref:spotify-for-anova-slice) 10 randomly selected rows from `spotify_for_anova`
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(ref:spotify-for-anova-slice) Five randomly selected rows from `spotify_for_anova`
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```{r echo=FALSE}
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spotify_for_anova |>
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slice_sample(n = 10) |>
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slice_sample(n = 5) |>
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kbl(caption = "(ref:spotify-for-anova-slice)") |>
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kable_styling(
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font_size = ifelse(is_latex_output(), 8, 16),
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#### Hypothesis tests for the partial slopes {-}
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We can now conduct hypothesis tests for the partial slopes in multiple linear regression. We can use the permutation test to test the null hypothesis $H_0: \beta_i = 0$ versus the alternative hypothesis $H_A: \beta_i \neq 0$ for each of the partial slopes. Let's use a significance level of $\alpha = 0.05$.
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We can visualize the $p$-values in the null distribution by comparing them to the observed test statistics. We do this by adding a `shade_p_value()` layer to the `visualize()` function.
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We can now conduct hypothesis tests for the partial slopes in multiple linear regression. We can use the permutation test to test the null hypothesis $H_0: \beta_i = 0$ versus the alternative hypothesis $H_A: \beta_i \neq 0$ for each of the partial slopes. Let's use a significance level of $\alpha = 0.05$. We can visualize the $p$-values in the null distribution by comparing them to the observed test statistics. We do this by adding a `shade_p_value()` layer `visualize()`.
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```{r, fig.height=ifelse(knitr::is_latex_output(), 6, 8), fig.width=6, fig.cap="Shaded p-values for the partial slopes in this multiple linear regression."}
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visualize(null_distribution_mlr) +

‎ModernDive.bbl

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\begin{thebibliography}{}
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\newblock {\em infer: Tidy Statistical Inference}.
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\newblock R package version 1.0.7.9000.
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\bibitem[Chernick and LaBudde, 2011]{Chernick2011}
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Chernick, M.~R. and LaBudde, R.~A. (2011).
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\newblock R package version 2.2.0.
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\newblock {\em Getting Used to R, {RStudio}, and R Markdown}.
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\bibitem[Kim and Ismay, 2024]{R-moderndive}
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Kim, A.~Y. and Ismay, C. (2024).
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\newblock {\em moderndive: Tidyverse-Friendly Introductory Linear Regression}.
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\newblock R package version 0.7.0.
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\newblock {\em ggplot2: Create Elegant Data Visualisations Using the Grammar of Graphics}.
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\newblock R package version 3.5.1.
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\bibitem[Wickham et~al., 2023]{R-dplyr}
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Wickham, H., François, R., Henry, L., Müller, K., and Vaughan, D. (2023).
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\newblock {\em dplyr: A Grammar of Data Manipulation}.
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\newblock R package version 1.1.4.
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\bibitem[Wickham et~al., 2024b]{R-readr}
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Wickham, H., Hester, J., and Bryan, J. (2024b).
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\newblock {\em readr: Read Rectangular Text Data}.
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\newblock R package version 2.1.5.
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\bibitem[Wickham et~al., 2024c]{R-tidyr}
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\end{thebibliography}

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