Simple Linear Regression
Sharon Kalondu 151929
2025-05-21
if (!"pacman" %in% installed.packages()[, "Package"]) {
install.packages("pacman", dependencies = TRUE)
library("pacman", character.only = TRUE)
}
pacman::p_load("here")
knitr::opts_knit$set(root.dir = here::here())1 Load the Dataset
1.0.1 Initial EDA
pacman::p_load("readr")
clv_data <- read_csv("./data/clv_data.csv")## Rows: 500 Columns: 2
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## dbl (2): purchase_frequency, customer_lifetime_value
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.head(clv_data)## # A tibble: 6 × 2
## purchase_frequency customer_lifetime_value
## <dbl> <dbl>
## 1 3 110.
## 2 7 190.
## 3 6 160.
## 4 2 94.4
## 5 4 133.
## 6 8 223.dim(clv_data)## [1] 500 2sapply(clv_data, class)## purchase_frequency customer_lifetime_value
## "numeric" "numeric"str(clv_data)## spc_tbl_ [500 × 2] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ purchase_frequency : num [1:500] 3 7 6 2 4 8 0 4 8 3 ...
## $ customer_lifetime_value: num [1:500] 110.3 190.2 160 94.4 133.2 ...
## - attr(*, "spec")=
## .. cols(
## .. purchase_frequency = col_double(),
## .. customer_lifetime_value = col_double()
## .. )
## - attr(*, "problems")=<externalptr>1.0.2 Measures of Central Tendency
summary(clv_data)## purchase_frequency customer_lifetime_value
## Min. :-1.000 Min. : 26.13
## 1st Qu.: 4.000 1st Qu.:122.04
## Median : 5.000 Median :148.21
## Mean : 4.914 Mean :148.25
## 3rd Qu.: 6.000 3rd Qu.:175.88
## Max. :11.000 Max. :262.041.0.2.1 Variance
sapply(clv_data[,], var) ## purchase_frequency customer_lifetime_value
## 4.146898 1642.3159961.0.2.2 Standard Deviation
sapply(clv_data[,], sd)## purchase_frequency customer_lifetime_value
## 2.036393 40.5254981.0.2.3 Kurtosis
pacman::p_load("e1071")
sapply(clv_data[,], kurtosis, type = 2)## purchase_frequency customer_lifetime_value
## -0.1220038 -0.14848111.0.2.4 Skewness
sapply(clv_data[,], skewness, type = 2)## purchase_frequency customer_lifetime_value
## -0.04021915 -0.016082421.1 Measures of Relationship
1.1.0.1 Covariance
cov(clv_data, method = "spearman")## purchase_frequency customer_lifetime_value
## purchase_frequency 20409.91 20235.73
## customer_lifetime_value 20235.73 20874.991.1.0.2 Correlation
cor.test(clv_data$customer_lifetime_value, clv_data$purchase_frequency, method = "spearman")##
## Spearman's rank correlation rho
##
## data: clv_data$customer_lifetime_value and clv_data$purchase_frequency
## S = 409190, p-value < 2.2e-16
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
## rho
## 0.9803588cor(clv_data, method = "spearman")## purchase_frequency customer_lifetime_value
## purchase_frequency 1.0000000 0.9803588
## customer_lifetime_value 0.9803588 1.00000001.2 Basic Visualization
1.2.0.1 Histogram
# `par(mfrow = c(1, 2))` This is used to divide the area used to plot the visualization into a 1 row by 2 columns grid
# `for (i in 1:2)` This is used to identify the variable (column) that is being processed
# `clv_data[[i]]` This is used to extract the i-th column as a vector
# `hist()` This is the function used to plot the histogram
par(mfrow = c(1, 2))
for (i in 1:2) {
if (is.numeric(clv_data[[i]])) {
hist(clv_data[[i]],
main = names(clv_data)[i],
xlab = names(clv_data)[i])
} else {
message(paste("Column", names(clv_data)[i], "is not numeric and will be skipped."))
}
}
1.2.0.2 Box and Whisker Plot
# `boxplot()` This is the function used to plot the box and whisker plot visualization
par(mfrow = c(1, 2))
for (i in 1:2) {
if (is.numeric(clv_data[[i]])) {
boxplot(clv_data[[i]], main = names(clv_data)[i])
} else {
message(paste("Column", names(clv_data)[i], "is not numeric and will be skipped."))
}
}
1.2.0.3 Missing Data Plot
pacman::p_load("Amelia")
missmap(clv_data, col = c("red", "grey"), legend = TRUE)
1.2.0.4 Correlation Plot
pacman::p_load("ggcorrplot")
ggcorrplot(cor(clv_data[,]))
1.2.0.5 Scatter Plot
pacman::p_load("corrplot")
pairs(clv_data$customer_lifetime_value ~ ., data = clv_data, col = clv_data$customer_lifetime_value)
pacman::p_load("ggplot2")
ggplot(clv_data,
aes(x = purchase_frequency, y = customer_lifetime_value)) +
geom_point() +
geom_smooth(method = lm) +
labs(
title = "Relationship between Customer Lifetime Value and Purchase Frequency",
x = "Purchase Frequency",
y = "Customer Lifetime Value"
)
1.2.1 Statistical Test
slr_test <-lm(customer_lifetime_value~purchase_frequency, data = clv_data)
summary(slr_test)##
## Call:
## lm(formula = customer_lifetime_value ~ purchase_frequency, data = clv_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.1176 -5.6169 -0.0491 5.6618 20.4837
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 52.2538 0.9042 57.79 <2e-16 ***
## purchase_frequency 19.5356 0.1700 114.91 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.734 on 498 degrees of freedom
## Multiple R-squared: 0.9637, Adjusted R-squared: 0.9636
## F-statistic: 1.32e+04 on 1 and 498 DF, p-value: < 2.2e-16confint(slr_test, level = 0.95)## 2.5 % 97.5 %
## (Intercept) 50.47731 54.03036
## purchase_frequency 19.20159 19.869651.3 Diagnostic EDA
1.3.0.1 Test of Linearity
plot(slr_test, which = 1)
1.3.0.2 Test of Independence of Errors
pacman::p_load("lmtest")
dwtest(slr_test)##
## Durbin-Watson test
##
## data: slr_test
## DW = 1.9104, p-value = 0.1573
## alternative hypothesis: true autocorrelation is greater than 01.3.0.3 Test of Normality
plot(slr_test, which = 2)
1.3.0.4 Test of Homoscedasticity
plot(slr_test, which = 3)
pacman::p_load("gvlma")
gvlma_results <- gvlma(slr_test)
summary(gvlma_results)##
## Call:
## lm(formula = customer_lifetime_value ~ purchase_frequency, data = clv_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -19.1176 -5.6169 -0.0491 5.6618 20.4837
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 52.2538 0.9042 57.79 <2e-16 ***
## purchase_frequency 19.5356 0.1700 114.91 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.734 on 498 degrees of freedom
## Multiple R-squared: 0.9637, Adjusted R-squared: 0.9636
## F-statistic: 1.32e+04 on 1 and 498 DF, p-value: < 2.2e-16
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = slr_test)
##
## Value p-value Decision
## Global Stat 5.08943 0.27824 Assumptions acceptable.
## Skewness 0.03973 0.84201 Assumptions acceptable.
## Kurtosis 3.61252 0.05735 Assumptions acceptable.
## Link Function 0.01459 0.90385 Assumptions acceptable.
## Heteroscedasticity 1.42258 0.23298 Assumptions acceptable.1.4 Interpretation of the Results
We can interpret the results of the statistical test with more confidence if the tests of assumptions are successful.
1.4.1 Academic Statement
A simple linear regression was conducted to examine the relationship between customer lifetime value (CLV) and purchase frequency. The results indicated that purchase frequency significantly predicted CLV, β = 19.54, 95% CI[19.20, 19.87], SE = 0.17, t(498) = 114.91, p < .001. The model explained 96.37% of the variance in CLV (R2 = .96, F(1, 498) = 13,200, p < .001). For every unit increase in purchase frequency, CLV increased by approximately 19.54 units. The intercept was 52.25, 95 % CI [50.48, 54.03], and the residual standard error was 7.73, indicating strong predictive accuracy.
1.4.2 Business Analysis
The strength of the relationship highlights the critical importance of customer retention. Initiatives that effectively encourage repeat purchases appear to be a primary driver of customer lifetime value based on this analysis. This understanding can guide the allocation of resources towards strategies that foster customer loyalty and encourage repeat business.
1.4.3 Limitations
The model employed is a simple linear regression, which only considers the linear relationship between purchase frequency and CLV. Other potentially influential factors that are not included in this model could also play a significant role in determining CLV, e.g., the average monetary value of each purchase.