Description

Below is a table of the with columns number, name, and a description:

senic <- read.csv("data/senic.csv") ## make sure you check your directory and that this file 

## Load description dataset
senic_description <- read.csv("Data/senic_description.csv")

## Demonstrate data.frame
senic_description.table <- xtable(senic_description)
print(senic_description.table, type="html")

SENIC Data Visualization

Here is the first 6 observations of the senic data:

senic.table <- xtable(head(senic))
print(senic.table, type="html")

I have also provided a correlation matrix to see which values are most closely associated:

senic_corr <- senic[-1][-7][-7]
corr.table <- xtable(cor(senic_corr))
print(corr.table, type="html")

QQ Plot of infection probability

ggqqplot(senic$infprob)

Here is a correlation matrix using corrplot package:

corrplot(cor(senic_corr), method = 'shade', order = 'AOE', diag = FALSE)

We can see from the data that there are several highly correlated variables. From here we can state our hypothesis and test the data so we can accept or reject the hypothesis.

Objective

To test whether the mean infection risk is the same in four geographic regions.

Action

1. Perform one-way ANOVA to compare mean infection risk across regions.
2. Conduct Tukey’s procedure to obtain confidence intervals for pairwise comparisons.
3. Perform a different pairwise comparison procedure (e.g., Bonferroni or Scheffé).

Result

senic$region <- factor(senic$region)
senic$medschl <- factor(senic$medschl)

model_a <- lm(infprob~region, data = senic)

anova(model_a)

## Analysis of Variance Table
## 
## Response: infprob
##            Df  Sum Sq Mean Sq F value  Pr(>F)  
## region      3  13.997  4.6656   2.714 0.04839 *
## Residuals 109 187.383  1.7191                  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Test whether infection probability is different among regions.

H0: region1=region2=region3=region4

H1: at least one region is different

If F* < F(0.95;3,109) = 3.239 accept H0, otherwise reject H0 and accept H1.

Since F*=2.714 we accept H0, which means that there is no significant difference between the regions at the 95% confidence level.

comp <- rbind('1' = c(1,0,0,0),
              '2' = c(0,1,0,0),
              '3' = c(0,0,1,0),
              '4' = c(0,0,0,1))

comp_test <- glht(model_a, linfct =  mcp(region=comp))
comp_tukey <- glht(model_a, linfct = mcp(region="Tukey"))

comp_test

## 
##   General Linear Hypotheses
## 
## Multiple Comparisons of Means: User-defined Contrasts
## 
## 
## Linear Hypotheses:
##        Estimate
## 1 == 0   0.0000
## 2 == 0  -0.4670
## 3 == 0  -0.9337
## 4 == 0  -0.4795

comp_tukey

## 
##   General Linear Hypotheses
## 
## Multiple Comparisons of Means: Tukey Contrasts
## 
## 
## Linear Hypotheses:
##            Estimate
## 2 - 1 == 0  -0.4670
## 3 - 1 == 0  -0.9337
## 4 - 1 == 0  -0.4795
## 3 - 2 == 0  -0.4667
## 4 - 2 == 0  -0.0125
## 4 - 3 == 0   0.4542

This data from Tukey tells us that the only significant difference is among region1 and region 3, since they are the only regions that don’t overlap with 0. This is more clear with the following plot.

mean_test <- emmeans(model_a, specs = "region")
mean_pairs <- pairs(mean_test)

tukey_pairs <- confint(mean_pairs, adjust="Tukey", level=0.95)
plot(tukey_pairs)

We will now perform the same pairwise comparisons using other procedures, Bonferrooni and Scheffe.

Bonferroni:

bonf_pairs <- confint(mean_pairs, adjust="bonferroni", level=0.95)
bonf_pairs

##  contrast          estimate    SE  df lower.CL upper.CL
##  region1 - region2   0.4670 0.339 109  -0.4448     1.38
##  region1 - region3   0.9337 0.328 109   0.0511     1.82
##  region1 - region4   0.4795 0.411 109  -0.6247     1.58
##  region2 - region3   0.4667 0.317 109  -0.3838     1.32
##  region2 - region4   0.0125 0.401 109  -1.0663     1.09
##  region3 - region4  -0.4542 0.392 109  -1.5085     0.60
## 
## Confidence level used: 0.95 
## Conf-level adjustment: bonferroni method for 6 estimates

region1 and region3 are the only significant pairs that don’t overlap with 0.

plot(bonf_pairs)

Scheffe:

scheffe_pairs <- confint(mean_pairs, adjust="scheffe", level=0.95)
scheffe_pairs

##  contrast          estimate    SE  df lower.CL upper.CL
##  region1 - region2   0.4670 0.339 109 -0.49651     1.43
##  region1 - region3   0.9337 0.328 109  0.00109     1.87
##  region1 - region4   0.4795 0.411 109 -0.68736     1.65
##  region2 - region3   0.4667 0.317 109 -0.43209     1.37
##  region2 - region4   0.0125 0.401 109 -1.12750     1.15
##  region3 - region4  -0.4542 0.392 109 -1.56825     0.66
## 
## Confidence level used: 0.95 
## Conf-level adjustment: scheffe method with rank 3

region1 and region3 are the only significant pairs that don’t overlap with 0.

plot(scheffe_pairs)

Conclusion

In this analysis, we aimed to assess whether there is a significant difference in infection probability (infprob) among four geographic regions (region1, region2, region3, region4) based on data from the SENIC project. We followed a step-by-step approach, including ANOVA, Tukey’s procedure, and alternative pairwise comparison procedures, to draw meaningful conclusions regarding regional differences in infection risk.

ANOVA: We initially conducted an analysis of variance (ANOVA) to test the null hypothesis (H0) that there is no significant difference in infection probability across the four geographic regions, versus the alternative hypothesis (H1) that at least one region differs from the others. Our ANOVA test yielded an F-statistic of F* = 2.714. We compared this statistic to the critical F(0.95;3,109) value, which is 3.239. Since F* < F(0.95;3,109), we accept the null hypothesis (H0), indicating that there is no significant difference between the regions at the 95% confidence level.

Post-hoc Pairwise Comparisons: To delve further into the differences among the regions, we performed pairwise comparisons using Tukey’s procedure, as well as alternative procedures such as Bonferroni and Scheffe. Here are the key findings:

• Tukey’s Procedure: After applying Tukey’s procedure, we found that the only significant mean comparison was between region1 and region3. All other pairwise comparisons were not significant.

• Bonferroni Procedure: The Bonferroni procedure also indicated that the only significant pairwise comparison was between region1 and region3.

• Scheffe Procedure: The Scheffe procedure yielded results consistent with Tukey and Bonferroni, identifying region1-region3 as the only significant pairwise comparison.