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Arthur DANJOU
03bf0a4db2
- Updated comments and code formatting in `3-td_ggplot2 - enonce.Rmd` for clarity. - Enhanced code structure in `4-td_graphiques - enonce.Rmd` by organizing options and library calls. - Replaced pipe operator `%>%` with `|>` in `Code_Lec3.Rmd` for consistency with modern R syntax. - Cleaned up commented-out code and ensured consistent spacing in ggplot calls.
140 lines
3.1 KiB
Plaintext
140 lines
3.1 KiB
Plaintext
```{r}
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setwd("/Users/arthurdanjou/Workspace/studies/M1/General Linear Models/TP1-bis")
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library(tidyverse)
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options(scipen = 999, digits = 5)
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```
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```{r}
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data <- read.csv("data01.csv", header = TRUE, sep = ",", dec = ".")
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head(data, 20)
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```
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```{r}
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ggplot(data, aes(x = cholesterol)) +
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geom_histogram(binwidth = 5, color = "black", fill = "gray80") +
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labs(x = "Cholesterol", y = "Frequency", title = "Histogram of cholesterol") +
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theme_bw(14)
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ggplot(data, aes(x = poids)) +
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geom_histogram(binwidth = 2.5, color = "black", fill = "gray80") +
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labs(x = "Poids", y = "Frequency", title = "Histogram of Poids") +
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theme_bw(14)
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ggplot(aes(y = cholesterol, x = poids), data = data) +
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geom_point() +
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labs(y = "Cholesterol (y)", x = "Poids, kg (x)", title = "Scatter plot of cholesterol and poids") +
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theme_bw(14)
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```
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# OLSE
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```{r}
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x <- data[, "poids"]
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y <- data[, "cholesterol"]
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Sxy <- sum((x - mean(x)) * y)
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Sxx <- sum((x - mean(x))^2)
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beta1 <- Sxy / Sxx
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beta0 <- mean(y) - beta1 * mean(x)
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c(beta0, beta1)
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```
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Final Equation: y = 18.4470 + 2.0523 * x
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```{r}
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X <- cbind(1, x)
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colnames(X) <- NULL
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X_t_X <- t(X) %*% X
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inv_X_t_x <- solve(X_t_X)
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betas <- inv_X_t_x %*% t(X) %*% y
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betas
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```
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```{r}
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model <- lm(data, formula = cholesterol ~ poids)
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summary(model)
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coef(model)
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```
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```{r}
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data <- data |>
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mutate(yhat = beta0 + beta1 * poids) |>
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mutate(residuals = cholesterol - yhat)
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data
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ggplot(data, aes(x = poids, y = cholesterol)) +
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geom_point(size = 2, shape = 21, fill = "blue", color = "cyan") +
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geom_line(aes(y = yhat), color = "blue") +
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labs(x = "Poids", y = "Cholesterol", title = "OLS Regression Line") +
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theme_bw(14)
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```
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```{r}
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mean(data[, "cholesterol"])
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mean(data[, "yhat"])
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mean(data[, "residuals"]) |> round(10)
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cov(data[, "residuals"], data[, "poids"]) |> round(10)
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(RSS <- sum((data[, "residuals"])^2))
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(TSS <- sum((y - mean(y))^2))
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TSS - beta1 * Sxy
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```
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```{r}
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dof <- nrow(data) - 2
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sigma_hat_2 <- RSS / dof
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cov_beta <- sigma_hat_2 * inv_X_t_x
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var_beta <- diag(cov_beta)
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std_beta <- sqrt(var_beta)
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sigma(model)^2
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vcov(model)
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```
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# Hypothesis Testing
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```{r}
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(t_beta <- beta1 / std_beta[2])
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qt(0.975, dof)
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2 * pt(abs(t_beta), dof, lower.tail = FALSE)
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summary(model)
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```
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```{r}
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MSS <- (TSS - RSS)
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(F <- MSS / (RSS / dof))
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qf(0.95, 1, dof)
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pf(F, 1, dof, lower.tail = FALSE)
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anova(model)
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```
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# Confidence Interval
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```{r}
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(CI_beta1 <- c(beta1 - qt(0.97, dof) * std_beta[2], beta1 + qt(0.97, dof) * std_beta[2]))
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confint(model, 0.95)
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t <- qt(0.975, dof)
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sigma_hat <- sigma(model)
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n <- nrow(data)
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data <- data |>
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mutate(error = t *
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sigma_hat *
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sqrt(1 / n + (poids - mean(poids))^2 / RSS)) |>
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mutate(conf.low = yhat - error, conf.high = yhat + error, error = NULL)
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ggplot(data, aes(x = poids, y = cholesterol)) +
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geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.3, fill = "blue") +
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geom_point(size = 2, shape = 21, fill = "blue", color = "cyan") +
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geom_line(aes(y = yhat), color = "blue") +
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labs(x = "Poids", y = "Cholesterol", title = "OLS Regression Line") +
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theme_bw(14)
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```
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# R^2
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```{r}
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(R2 <- MSS / TSS)
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summary(model)$r.squared
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cor(data[, "cholesterol"], data[, "poids"])^2
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```
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