Understanding regression
getwd()
## [1] "C:/Users/Dell/Downloads"
## Example: Space Shuttle Launch Data
launch <- read.csv("challenger2.csv")
# estimate beta manually
b <- cov(launch$temperature, launch$distress_ct) / var(launch$temperature)
b
## [1] -0.03364796
#A one-unit increase in temperature is associated with an average decrease of 0.03364796 units in the distress count.
# estimate alpha manually
a <- mean(launch$distress_ct) - b * mean(launch$temperature)
a
## [1] 2.814585
#Baseline Distress: This value suggests a baseline distress count of 2.814585, even when the temperature is zero. This means that even in ideal conditions (no temperature stress), some level of distress is expected
# calculate the correlation of launch data
r <- cov(launch$temperature, launch$distress_ct) /
(sd(launch$temperature) * sd(launch$distress_ct))
r
## [1] -0.3359996
#This correlation is less significant than in challenger.csv (51%). Moderate correlation. Meaning: the variable that we chose to explain distress events (temperature),explains distress in 33.59%.
cor(launch$temperature, launch$distress_ct)
## [1] -0.3359996
# computing the slope using correlation
r * (sd(launch$distress_ct) / sd(launch$temperature))
## [1] -0.03364796
# confirming the regression line using the lm function (not in text)
model <- lm(distress_ct ~ temperature, data = launch)
model
##
## Call:
## lm(formula = distress_ct ~ temperature, data = launch)
##
## Coefficients:
## (Intercept) temperature
## 2.81458 -0.03365
summary(model)
##
## Call:
## lm(formula = distress_ct ~ temperature, data = launch)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.0649 -0.4929 -0.2573 0.3052 1.7090
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.81458 1.24629 2.258 0.0322 *
## temperature -0.03365 0.01815 -1.854 0.0747 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7076 on 27 degrees of freedom
## Multiple R-squared: 0.1129, Adjusted R-squared: 0.08004
## F-statistic: 3.436 on 1 and 27 DF, p-value: 0.07474
# Indicates that only about 11% of the variability in distress count is explained by temperature. Other factors likely play a role, compared to first database where R-squared explained 26% of the variability in distress count.
# p-value of 0.0747 There might be a weak negative relationship between temperature and distress count. However, the evidence is not strong enough to conclude this definitively.
# creating a simple multiple regression function
reg <- function(y, x) {
x <- as.matrix(x)
x <- cbind(Intercept = 1, x)
b <- solve(t(x) %*% x) %*% t(x) %*% y
colnames(b) <- "estimate"
print(b)
}
# examine the launch data
str(launch)
## 'data.frame': 29 obs. of 4 variables:
## $ distress_ct : int 0 1 0 0 0 0 0 0 1 1 ...
## $ temperature : int 66 70 69 68 67 72 73 70 57 63 ...
## $ field_check_pressure: int 50 50 50 50 50 50 100 100 200 200 ...
## $ flight_num : int 1 2 3 4 5 6 7 8 9 10 ...
# test regression model with simple linear regression
reg(y = launch$distress_ct, x = launch[2])
## estimate
## Intercept 2.81458456
## temperature -0.03364796
# use regression model with multiple regression
reg(y = launch$distress_ct, x = launch[2:4])
## estimate
## Intercept 2.239817e+00
## temperature -3.124185e-02
## field_check_pressure -2.586765e-05
## flight_num 2.762455e-02
# confirming the multiple regression result using the lm function (not in text)
model <- lm(distress_ct ~ temperature + field_check_pressure + flight_num, data = launch)
model
##
## Call:
## lm(formula = distress_ct ~ temperature + field_check_pressure +
## flight_num, data = launch)
##
## Coefficients:
## (Intercept) temperature field_check_pressure
## 2.240e+00 -3.124e-02 -2.587e-05
## flight_num
## 2.762e-02