Creating the linear model to assess three-point percentage

# Creating the linear model 

lm_model <- lm(fg_3p ~ dist + x3p_percent_cor_3 + fga_3p, data = df)

summary(lm_model)
## 
## Call:
## lm(formula = fg_3p ~ dist + x3p_percent_cor_3 + fga_3p, data = df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.42971 -0.03101  0.00727  0.03930  0.48813 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        0.128739   0.006929  18.580  < 2e-16 ***
## dist               0.006486   0.001018   6.373 2.16e-10 ***
## x3p_percent_cor_3  0.355870   0.008521  41.762  < 2e-16 ***
## fga_3p            -0.052435   0.022507  -2.330   0.0199 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.08254 on 2806 degrees of freedom
##   (447 observations deleted due to missingness)
## Multiple R-squared:  0.4408, Adjusted R-squared:  0.4402 
## F-statistic: 737.2 on 3 and 2806 DF,  p-value: < 2.2e-16

Insights: The linear regression shows that all three variables are statistically significant in explaining the variation in overall three-point percentage. Of the variables, corner 3pt% (x3p_percent_cor_3) has the strongest effect with a positive coefficient of 0.356, meaning those with a higher accuracy from the corner tend to have higher 3pt% overall. It is important to note that there is a mechanical relationship between corner 3pt% and overall 3pt% since corner 3pt% is part of the overall 3pt%. Average shot distance (dist) also has a positive effect, meaning players who take longer shots tend to have a higher 3pt% since most of their shots are behind the three-point arc. Three-point attempt (fga_3p) has a small negative effect on 3pt%, meaning players who attempt more three-pointers tend to have a lower efficiency. Significance: The model explains a good portion of the variation in 3pt% (R^2 approximately 44%) indicating a strong association between these variables and 3pt%, though it does not imply a casual relationship. The results show that corner 3pt% is the most important factor in determining overall three-point shooting of these variables, meaning players who excel in high-efficiency shot types (corner 3s are considered the most efficient three-point shot due to the player being the closest to the basket on the three-point arc), tend to be strong shooters overall. The negative relationship between three-point attempts and three-point shooting is natural since players who take high-volume (more difficult) three-point shots usually have lower efficiency. Further Question: Would including a variable to account for shot quality/difficulty or defensive pressure improve the model?

Corner 3pt% Coefficient Interpretation

Interpretation: Holding dist and fga_3p constant, a 1 percentage point increase in corner three-point percentage is associated with an approximate 0.356 percentage point increase in overall three-point percentage. Insights: The coefficient for corner 3pt% (0.356) is large and significant, meaning a player’s corner three accuracy is strongly associated with their overall three-point shooting performance. Significance: This suggests that corner three-point shooting is a strong indicator of overall three-point shooting ability. With corner threes being the highest percentage shot behind the three-point line (due to it being the shortest distance from the basket), players who shoot well from the corner tend to be more efficient shooters overall. Further Question: Are these players specializing in just the corner three or do they shoot well overall from three regardless of spot?

Model Diagnostic (Residual vs Fitted)

# Residual vs Fitted
plot(lm_model, which = 1)

Insights: While the residuals are generally centered around zero, the red line shows a curve meaning the relationship between the independent variables and 3pt% in not linear. Issue/Severity: The curved trend suggests the model does not fully capture the true relationship between the variables. There is slight heteroskedasticity since the spread of residuals slightly increase as fitted values increase. This would be considered moderate severity leading to a moderate confidence in the linear model capturing this relationship. Significance: While the model captures the general trend in 3pt%, it most likely oversimplifies the relationship between shooting behavior and shot performance. The effects of corner 3pt% or shot volume may not be strictly linear across all ranges. Another perspective to consider is removing outliers with data cleaning, specifically low-volume players taking one shot for example as this is standard activity in the professional basketball analytics field.
Further Question: Would adding a non-linear variable improve the model’s fit?