Violent discipline

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summary(cars)

##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

## correlations between the mean wvs value (jvmean) and the percentage of "always justifiable" (jv0) with rates of violence (violence)
correlation <- cor(data$violence, data$jv0)
correlation

## [1] 0.1266883

correlation <- cor(data$violence, data$jvmean)
correlation

## [1] 0.2218276

## scatterplot shows no patterns, so we try to test a non-linear relationship with a quadratic polynomial regression

lm_poly <- lm(jv0 ~ poly(violence, 2), data = data)
summary(lm_poly)

## 
## Call:
## lm(formula = jv0 ~ poly(violence, 2), data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.7626 -2.4869 -0.5565  1.2595 11.9352 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          3.7729     0.8507   4.435  0.00023 ***
## poly(violence, 2)1   2.8771     4.1675   0.690  0.49753    
## poly(violence, 2)2  11.9467     4.1675   2.867  0.00924 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.168 on 21 degrees of freedom
## Multiple R-squared:  0.2928, Adjusted R-squared:  0.2254 
## F-statistic: 4.347 on 2 and 21 DF,  p-value: 0.02632

lm_poly <- lm(jvmean ~ poly(violence, 2), data = data)
summary(lm_poly)

## 
## Call:
## lm(formula = jvmean ~ poly(violence, 2), data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.0222 -0.5311 -0.2489  0.3395  2.0927 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          2.9483     0.1952  15.104 9.42e-13 ***
## poly(violence, 2)1   1.0534     0.9563   1.102    0.283    
## poly(violence, 2)2   1.4951     0.9563   1.563    0.133    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9563 on 21 degrees of freedom
## Multiple R-squared:  0.1483, Adjusted R-squared:  0.06722 
## F-statistic: 1.829 on 2 and 21 DF,  p-value: 0.1853

## non-parametric correlation with Spearman's

cor.test(data$violence, data$jvmean, method = "spearman")

## 
##  Spearman's rank correlation rho
## 
## data:  data$violence and data$jvmean
## S = 2304, p-value = 0.9951
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##         rho 
## -0.00173913

cor.test(data$violence, data$jv0, method = "spearman")

## Warning in cor.test.default(data$violence, data$jv0, method = "spearman"): Kann
## exakten p-Wert bei Bindungen nicht berechnen

## 
##  Spearman's rank correlation rho
## 
## data:  data$violence and data$jv0
## S = 2422.2, p-value = 0.8053
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##         rho 
## -0.05312438

## lets try to remove outliers

grubbs.test(data$jvmean)

## 
##  Grubbs test for one outlier
## 
## data:  data$jvmean
## G = 3.09211, U = 0.56622, p-value = 0.005598
## alternative hypothesis: highest value 6.01 is an outlier

grubbs.test(data$violence)

## 
##  Grubbs test for one outlier
## 
## data:  data$violence
## G = 1.86208, U = 0.84269, p-value = 0.66
## alternative hypothesis: highest value 90.4 is an outlier

data <- data[-15, ]

## with outlier excluded, lets try correlation again

cor.test(data$violence, data$jvmean, method = "spearman")

## 
##  Spearman's rank correlation rho
## 
## data:  data$violence and data$jvmean
## S = 2304, p-value = 0.5274
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##        rho 
## -0.1383399

cor.test(data$violence, data$jv0, method = "spearman")

## Warning in cor.test.default(data$violence, data$jv0, method = "spearman"): Kann
## exakten p-Wert bei Bindungen nicht berechnen

## 
##  Spearman's rank correlation rho
## 
## data:  data$violence and data$jv0
## S = 2422.7, p-value = 0.3677
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##        rho 
## -0.1969812

## pvalue lower but still not significant

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