Multiple linear regression estimates how a continuous target changes with two or more predictors. It produces a straight-line equation through the data, helping quantify the predictors’ collective relationship with variation in the target. A simple linear regression uses only one predictor; predictors are usually continuous but may be categorical. section 5.0 section 5.7
Use it for questions such as: which NBA hustle statistics are associated with regular-season wins, and by how much? Here, wins are modeled from measures including screen-assist points, deflections, and loose balls recovered. The aim is not merely prediction; it can also be isolating predictors with statistically significant relationships to an outcome. section 5.0 section 20.4
A practical linear-regression workflow is:
- Explore and wrangle the data first; identify and treat outliers, assess distributions, and examine correlations to find promising predictors. section 5.9 section 20.4
- Split observations into mutually exclusive training and prediction subsets to reduce overfitting. section 5.7 section 5.9
- Fit the model, inspect returned estimates and diagnostics, and check predictors for multicollinearity. section 5.7
- Compare competing predictor sets using adjusted R², AIC, significance, and diagnostic results rather than trusting one fit automatically. section 5.7.6
Reducing a wins model
A reduced R model fits wins from screen-assist points, deflections, and loose balls. Each displayed predictor has a p-value below 0.05. Compared with the larger model, it has adjusted R² of 0.16 rather than 0.14 and AIC of 514 rather than 518—an improvement, though it still explains limited variation in wins. section 5.7.6 section 5.9
fit2 <- lm(wins ~ screen_assists_pts + deflections + loose_balls,
data = train)
tidy(fit2)
## # A tibble: 4 × 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -56.1 25.5 -2.20 0.0317
## 2 screen_assists_pts 1.12 0.422 2.65 0.0101
## 3 deflections 2.35 0.859 2.74 0.00805
## 4 loose_balls 4.81 1.98 2.42 0.0182
augment(fit2) -> fit_tbl2
print(fit_tbl2)
# A tibble: 67 × 10
## wins screen_assists_pts deflections loose_balls .fitted .resid
## <int> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 49 20 14.1 8.3 39.3 9.66
## 2 42 26.2 12.1 8 40.1 1.85
## 3 39 25.7 12.6 8.1 41.2 -2.24
## 4 19 22.4 11.8 7.6 33.3 -14.3
## 5 33 20.1 11.5 8.4 33.8 -0.825
## 6 54 19.9 13.9 8.6 40.2 13.8
## 7 57 26.3 14.2 8.7 48.6 8.44
## 8 53 16.6 14.9 8.4 37.9 15.1
## 9 48 20.2 14.2 8.6 41.2 6.76
##10 37 20.1 11.9 8.5 35.2 1.75
## hat .sigma .cooksd .std.resid
## <dbl> <dbl> <dbl> <dbl>
## 1 0.0224 10.8 0.00472 0.907
## 2 0.0726 10.8 0.000626 0.179
## 3 0.0568 10.8 0.000691 -0.214
## 4 0.0647 10.7 0.0324 -1.37
## 5 0.0749 10.9 0.000128 -0.0797
## 6 0.0316 10.7 0.0138 1.30
## 7 0.0792 10.8 0.0144 0.817
## 8 0.0550 10.7 0.0303 1.44
## 9 0.0298 10.8 0.00312 0.637
##10 0.0632 10.9 0.000478 0.168
## # ... with 57 more rows
fit_tbl2 %>%
mutate(wins_dif = abs(wins - .fitted)) -> fit_tbl2
mean(fit_tbl2$wins_dif)
## [1] 8.427093
glance(fit2)
## # A tibble: 1 × 12
## r.squared adj.r.squared sigma statistic p.value df logLik
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.194 0.156 10.8 5.06 0.00334 3 -252.
## AIC BIC deviance df.residual nobs
## dbl> <dbl> <dbl> <int> <int>
## 1 514. 525. 7302. 63 67
vif(fit2)
## screen_assists_pts deflections loose_balls
## 1.085184 1.031128 1.098619
par(mfrow = c(2,2))
plot(fit2)
Inspect an executable lm() fit, coefficient table, fitted values, and residuals for the reduced wins model. section 5.7.6
A regression line minimizes differences between the line and observed data. For a quick visual view of a two-variable relationship, R can add an lm trend line to a scatterplot; this shows direction and pattern but does not replace the full multiple-regression workflow. section 4.5.3 section 5.7
Choose linear regression when the target is continuous. For a binary target, such as whether a team finishes with a winning record, use logistic regression instead. Also remember that real data may not be linear, so modest fit can still support a narrower finding about which predictors matter. section 5.7 section 5.9
