Data Dive Week - 8
Anurag
2024-03-03
Contents:
1. ANOVA - Finding the Runs Scored by teams in the three divisions (East, West, Central)
2. Linear Regression Model - Runs Scored and Batting Average of teams.
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## somelahman_data = read.csv("/Users/anuragreddy/Desktop/Statistics with R/Lahmans Databse .csv")Div_df <- lahman_data |>
filter(yearID != 2020) |>
select(franchID, divID, R) |>
group_by(divID)
Div_df## # A tibble: 630 × 3
## # Groups: divID [3]
## franchID divID R
## <chr> <chr> <int>
## 1 ANA W 864
## 2 BAL E 794
## 3 BOS E 792
## 4 CHW C 978
## 5 CLE C 950
## 6 DET C 823
## 7 KCR C 879
## 8 MIN C 748
## 9 NYY E 871
## 10 OAK W 947
## # ℹ 620 more rowsNull Hypothesis: There is no difference between the runs scored by teams in the all three divisions.
Alternative Hypothesis: There exists a significant difference in the runs scored by teams across the three divisions.
Div_df |>
ggplot(aes(x=R, fill = divID))+
geom_boxplot()+
labs(title = "Runs Distribution Division-wise", x= "Runs Scored")+
theme_economist() +
scale_colour_economist()
Div_df |>
group_by(divID)|>
summarise(Avg_Runs_Scored = mean(R))|>
select(divID,Avg_Runs_Scored)## # A tibble: 3 × 2
## divID Avg_Runs_Scored
## <chr> <dbl>
## 1 C 731.
## 2 E 746.
## 3 W 741.Interpretation: The distribution of runs scored exhibits a notable degree of uniformity across the three divisions. We are interested in determining whether this provides enough evidence against the alternative hypothesis stated above.
ANOVA Testing:
Div_Anova <- aov(R ~ divID, data = Div_df)
summary(Div_Anova)## Df Sum Sq Mean Sq F value Pr(>F)
## divID 2 24041 12021 1.75 0.175
## Residuals 627 4307245 6870Interpretation: The analysis of variance (ANOVA) results indicate that there are two sources of variation considered in this study: the “divID” factor and the residuals. The “divID” factor has 2 degrees of freedom, with a sum of squares of 24041 and a mean square of 12021. The F value associated with this factor is 1.75, and the corresponding p-value is 0.175.
The non-significant p-value (0.175>0.05 (sig value) suggests that there is insufficient evidence to reject the null hypothesis. Consequently, we do not have enough statistical evidence to conclude that there is a significant difference in the runs scored by teams across the three divisions based on the “divID” factor.
pairwise.t.test(Div_df$R, Div_df$divID, p.adjust.method = "bonferroni")##
## Pairwise comparisons using t tests with pooled SD
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## data: Div_df$R and Div_df$divID
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## C E
## E 0.20 -
## W 0.75 1.00
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## P value adjustment method: bonferroniInterpretation: There is not enough statistical evidence to conclude that there is a significant difference in the runs scored by teams between the three divisions based on the “divID” factor.
2. Linear Regression - Runs Scored (R) ~ Batting Average
[BA - Hits (H)/At Bats (AB)]
Runs_BA <- lahman_data |>
filter(yearID !=2020) |>
mutate(BA = H/AB) |>
select(R,BA)
head(Runs_BA)## R BA
## 1 864 0.2796731
## 2 794 0.2717607
## 3 792 0.2669627
## 4 978 0.2860432
## 5 950 0.2884040
## 6 823 0.2751595Runs_BA |>
ggplot(aes(x=BA, y=R))+
geom_point()+
geom_smooth(method = "lm", se = FALSE, color = 'darkblue')+
labs(title = "Runs Scored vs Batting Average", x = "Batting Average", y = "Runs Scored", color='darkblue')+
theme_solarized()## `geom_smooth()` using formula = 'y ~ x'
ASSUMPTION 1: VARIABLE \(x\) IS LINEARLY CORRELATED WITH RESPONSE \(y\).
Interpretation: Based on the aforementioned visualization, one can deduce the presence of a linear correlation between the response variable, runs scored, and the explanatory variable, batting average.
ASSUMPTION 2: ERRORS HAVE CONSTANT VARIANCE ACROSS ALL PREDICTIONS
ASSUMPTION 3: OBSERVATIONS ARE INDEPENDENT AND UNCORRELATED
linear_model <- lm(R~BA,Runs_BA)
linear_model##
## Call:
## lm(formula = R ~ BA, data = Runs_BA)
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## Coefficients:
## (Intercept) BA
## -477.9 4708.9Interpretation:
- Intercept (-477.9): The intercept represents the estimated value of the response variable (runs scored, denoted as R) when the explanatory variable (batting average, denoted as BA) is zero. However, in the context of batting average, this value might not have a practical interpretation, as a batting average of zero is not meaningful.
- BA (Batting Average, 4708.9): The coefficient for batting average (BA) is 4708.9. This implies that, on average, for each unit increase in batting average, the model predicts an increase of 4708.9 units in runs scored. This positive coefficient suggests a positive linear relationship between batting average and runs scored: as the batting average goes up, the predicted runs scored also increase.