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Linear regression implementation using linear algebra in R
### Linear Regression Using lm() ----------------------------------------
data("swiss")
dat <- swiss
linear_model <- lm(Fertility ~ ., data = dat)
summary(linear_model)
# Call:
# lm(formula = Fertility ~ ., data = dat)
#
# Residuals:
# Min 1Q Median 3Q Max
# -15.2743 -5.2617 0.5032 4.1198 15.3213
#
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 66.91518 10.70604 6.250 1.91e-07 ***
# Agriculture -0.17211 0.07030 -2.448 0.01873 *
# Examination -0.25801 0.25388 -1.016 0.31546
# Education -0.87094 0.18303 -4.758 2.43e-05 ***
# Catholic 0.10412 0.03526 2.953 0.00519 **
# Infant.Mortality 1.07705 0.38172 2.822 0.00734 **
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 7.165 on 41 degrees of freedom
# Multiple R-squared: 0.7067, Adjusted R-squared: 0.671
# F-statistic: 19.76 on 5 and 41 DF, p-value: 5.594e-10
### Using Linear Algebra ------------------------------------------------
y <- matrix(dat$Fertility, nrow = nrow(dat))
X <- cbind(1, as.matrix(x = dat[,-1]))
colnames(X)[1] <- "(Intercept)"
# N x k matrix
N <- nrow(X)
k <- ncol(X) - 1 # number of predictor variables (ergo, excluding Intercept column)
# Estimated Regression Coefficients
beta_hat <- solve(t(X)%*%X)%*%(t(X)%*%y)
# Variance of outcome variable = Variance of residuals
sigma_sq <- residual_variance <- (N-k-1)^-1 * sum((y - X %*% beta_hat)^2)
residual_std_error <- sqrt(residual_variance)
# Variance and Std. Error of estimated coefficients of the linear model
var_betaHat <- sigma_sq * solve(t(X) %*% X)
coeff_std_errors <- sqrt(diag(var_betaHat))
# t values of estimates are ratio of estimated coefficients to std. errors
t_values <- beta_hat / coeff_std_errors
# p-values of t-statistics of estimated coefficeints
p_values_tstat <- 2 * pt(abs(t_values), N-k, lower.tail = FALSE)
# assigning R's significance codes to obtained p-values
signif_codes_match <- function(x){
ifelse(x <= 0.001,"***",
ifelse(x <= 0.01,"**",
ifelse(x < 0.05,"*",
ifelse(x < 0.1,"."," "))))
}
signif_codes <- sapply(p_values_tstat, signif_codes_match)
# R-squared and Adjusted R-squared (refer any econometrics / statistics textbook)
R_sq <- 1 - (N-k-1)*residual_variance / (N*mean((y - mean(y))^2))
R_sq_adj <- 1 - residual_variance / ((N/(N-1))*mean((y - mean(y))^2))
# Residual sum of squares (RSS) for the full model
RSS <- (N-k-1)*residual_variance
# RSS for the partial model with only intercept (equal to mean), ergo, TSS
RSS0 <- TSS <- sum((y - mean(y))^2)
# F statistic based on RSS for full and partial models
# k = degress of freedom of partial model
# N - k - 1 = degress of freedom of full model
F_stat <- ((RSS0 - RSS)/k) / (RSS/(N-k-1))
# p-values of the F statistic
p_value_F_stat <- pf(F_stat, df1 = k, df2 = N-k-1, lower.tail = FALSE)
# stitch the main results toghether
lm_results <- as.data.frame(cbind(beta_hat, coeff_std_errors,
t_values, p_values_tstat, signif_codes))
colnames(lm_results) <- c("Estimate","Std. Error","t value","Pr(>|t|)","")
### Print out results of all relevant calcualtions -----------------------
print(lm_results)
cat("Residual standard error: ",
round(residual_std_error, digits = 3),
" on ",N-k-1," degrees of freedom",
"\nMultiple R-squared: ",R_sq," Adjusted R-squared: ",R_sq_adj,
"\nF-statistic: ",F_stat, " on ",k-1," and ",N-k-1,
" DF, p-value: ", p_value_F_stat,"\n")
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 66.9151816789654 10.7060375853301 6.25022854119771 1.73336561301153e-07 ***
# Agriculture -0.172113970941457 0.0703039231786469 -2.44814177018405 0.0186186100433133 *
# Examination -0.258008239834722 0.253878200892098 -1.01626779663678 0.315320687313066
# Education -0.870940062939429 0.183028601571259 -4.75849159892283 2.3228265226988e-05 ***
# Catholic 0.104115330743766 0.035257852536169 2.95296858017545 0.00513556154915653 **
# Infant.Mortality 1.07704814069103 0.381719650858061 2.82156849475775 0.00726899472564356 **
# Residual standard error: 7.165 on 41 degrees of freedom
# Multiple R-squared: 0.706735 Adjusted R-squared: 0.670971
# F-statistic: 19.76106 on 4 and 41 DF, p-value: 5.593799e-10
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