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Hypothesis Testing II

Zhaoxia Yu

Load packages, read data

Code

library(tidyverse)
library(ggplot2)
alzheimer_data <- read.csv('data/alzheimer_data.csv') %>% 
  select(id, diagnosis, age, educ, female, height, weight, lhippo, rhippo) %>% 
  mutate(diagnosis = as.factor(diagnosis), 
         female = as.factor(female),
         hippo=lhippo + rhippo)


alzheimer_healthy <- read.csv('data/alzheimer_data.csv') %>% 
  select(id, diagnosis, age, educ, female, height, weight, lhippo, rhippo) %>% 
  mutate(diagnosis = as.factor(diagnosis),
         female = as.factor(female), 
         hippo=lhippo+rhippo) %>% 
  filter(diagnosis==0)

A LIST OF QUESTIONS

Ex 1. Is the hippocampal volume of healthy males greater than 6 cm\(^3\)? (Done)
Ex 2. Is the proportion of healthy adults with a right hippocampal volume > 3cm\(^3\) equal to 50%? (can be done by using the method for testing a proportion: z-test)
Ex 3. Do healthy men and women have the same mean left hippocampal volume?
Ex 4. Is the left hippocampal volume equal to the right hippocampal volume in humans?
Ex 5. Is having a large hippocampus (> 7 cm\(^3\)) associated with gender?
Ex 6. Is hippocampal volume correlated with age in healthy adults?
Ex 7. Does left hippocampal volume differ across diagnostic groups?

EXAMPLE 3: Two-Sample t-test

Ex 3. Do healthy men and women have the same mean left hippocampal volume?

Categorical (gender) and Numerical (Left Hippocampus) :
- Side by Side Boxplot
- Side by Side Violin
- 2 Histograms

Example 3: Visualization – boxplot

Code

ggplot(data= alzheimer_healthy,
       mapping= aes(x= female, 
                    y=lhippo,
                    fill=female)) +
  geom_boxplot()+
  labs(
    title="Boxplot of lhippo by Gender",
    x="Gender",
    y="LHippo volume (cm^3)") +  
  theme_minimal()

Example 3: Visualization – violin

Code

ggplot(data= alzheimer_healthy,
       mapping= aes(x= female, 
                    y=lhippo,
                    fill=female)) +
  geom_violin()+
  labs(
    title="Violin of lhippo by Gender",
    x="Gender",
    y="LHippo volume (cm^3)") +  
  theme_minimal()

Example 3: Visualization – histogram

Code

ggplot(data= alzheimer_healthy,
       mapping= aes(x=lhippo,
                    fill=female)) +
  geom_histogram(bins=35)+
  labs(
    title="Histogram of Lhippo by Gender",
    x="LHippo volume (cm^3)",
    y="Count")

Example 3: use two-sample t-test

Code

t.test(alzheimer_healthy$lhippo[alzheimer_healthy$female==0], alzheimer_healthy$lhippo[alzheimer_healthy$female==1])


    Welch Two Sample t-test

data:  alzheimer_healthy$lhippo[alzheimer_healthy$female == 0] and alzheimer_healthy$lhippo[alzheimer_healthy$female == 1]
t = 11.756, df = 971.28, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.2088777 0.2925882
sample estimates:
mean of x mean of y 
 3.345749  3.095016

Code

#alternative way to do the same test
t.test(alzheimer_healthy$lhippo ~ alzheimer_healthy$female)


    Welch Two Sample t-test

data:  alzheimer_healthy$lhippo by alzheimer_healthy$female
t = 11.756, df = 971.28, p-value < 2.2e-16
alternative hypothesis: true difference in means between group 0 and group 1 is not equal to 0
95 percent confidence interval:
 0.2088777 0.2925882
sample estimates:
mean in group 0 mean in group 1 
       3.345749        3.095016

Example 4

Research Question: Is the left hippocampal volume equal to the right hippocampal volume in humans?
Null Hypothesis: Human’s left hippocampus volume is the same as the right hippocampus volume.
Alternative Hypothesis: Human’s left hippocampus volume is not the same as the right hippocampus volume.

\[H_0: \mu_L = \mu_R \mbox{ vs } \mu_L \not= \mu_R\]

Method: paired t-test

Example 4

It is attempting to perform a two-sample t-test
However, one fundamental assumption in the two-sample problem is that the two samples should be independent, such as the female group vs the male group.
Here we are looking at two features (left vs right hippocampus) of the same subjects. The correct test is
paired t-test, which equivalent to perform a one-sample t-test using the differences

Example 4

Code

t.test(alzheimer_data$rhippo, alzheimer_data$lhippo, paired = T)


    Paired t-test

data:  alzheimer_data$rhippo and alzheimer_data$lhippo
t = 17, df = 2699, p-value < 2.2e-16
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 0.07008463 0.08836085
sample estimates:
mean difference 
     0.07922274

Code

#equivalently, 
t.test(alzheimer_data$rhippo- alzheimer_data$lhippo)


    One Sample t-test

data:  alzheimer_data$rhippo - alzheimer_data$lhippo
t = 17, df = 2699, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
 0.07008463 0.08836085
sample estimates:
 mean of x 
0.07922274

Example 5

Research Question: Is having a large hippocampus (> 7 cm\(^3\)) associated with gender?
This is a category variable vs category variable problem
Method: chi-squared test

Code

table(alzheimer_data$hippo>7,
      alzheimer_data$female)

       
           0    1
  FALSE  843 1374
  TRUE   308  175

Code

chisq.test(table(alzheimer_data$rhippo>7,
      alzheimer_data$female))


    Chi-squared test for given probabilities

data:  table(alzheimer_data$rhippo > 7, alzheimer_data$female)
X-squared = 58.668, df = 1, p-value = 1.866e-14

Example 6

Research Question: hippocampal volume correlated with age in healthy adults?
Null Hypothesis: hippocampal volume and age are not correlated.
Alternative Hypothesis: hippocampal volume and age are not correlated.

\[H_0: \rho=0 \mbox{ vs } H_1: \rho\not=0\]

Method: correlation test, more general, linear regression.

Example 6: Visualization

Code

#plot(alzheimer_healthy$age, alzheimer_healthy$hippo)
ggplot(data= alzheimer_healthy,
       mapping= aes(x= age, 
                    y= hippo)) +
  geom_point() +
  labs(
    title="Scatter plot of hippocampal volume vs age",
    x="Age (years)",
    y="Hippocampal volume (cm^3)") +  
  theme_minimal()

Discussion: is there a linear trend? are there regions of concerns?

Example 6

Code

cor.test(alzheimer_healthy$age, alzheimer_healthy$hippo)


    Pearson's product-moment correlation

data:  alzheimer_healthy$age and alzheimer_healthy$hippo
t = -15.125, df = 1532, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.4032192 -0.3160971
sample estimates:
      cor 
-0.360444

Code

#alternatively, we can use linear regression
summary(lm(alzheimer_healthy$hippo ~ alzheimer_healthy$age))


Call:
lm(formula = alzheimer_healthy$hippo ~ alzheimer_healthy$age)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.57311 -0.48368 -0.01818  0.49369  2.80738 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)            8.00629    0.10570   75.75   <2e-16 ***
alzheimer_healthy$age -0.02329    0.00154  -15.12   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7209 on 1532 degrees of freedom
Multiple R-squared:  0.1299,    Adjusted R-squared:  0.1294 
F-statistic: 228.8 on 1 and 1532 DF,  p-value: < 2.2e-16

Example 7

Research Question: Does left hippocampal volume differ across diagnostic groups?
Null Hypothesis:
Alternative Hypothesis:

\[H_0: \mu_0=\mu_1=\mu_2 \mbox{ vs } H_1: \mbox{ at least two means are different}\]

Method: one-way analysis of variance (ANOVA), more generally, linear regression

Example 7

Code

ggplot(data= alzheimer_data,
       mapping= aes(x= diagnosis, 
                    y=lhippo,
                    fill=diagnosis)) +
  geom_boxplot()+
  labs(
    title="Boxplot of lhippo by Diagnosis",
    x="Diagnosis",
    y="LHippo volume (cm^3)") +  
  theme_minimal()

Code

summary(aov(alzheimer_data$lhippo ~ alzheimer_data$diagnosis))

                           Df Sum Sq Mean Sq F value Pr(>F)    
alzheimer_data$diagnosis    2  100.5   50.27   247.1 <2e-16 ***
Residuals                2697  548.6    0.20                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Code

#equivalently
summary(aov(lhippo ~ diagnosis, data=alzheimer_data))

              Df Sum Sq Mean Sq F value Pr(>F)    
diagnosis      2  100.5   50.27   247.1 <2e-16 ***
Residuals   2697  548.6    0.20                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Summary

Ex 1. Is the hippocampal volume of healthy males greater than 6 cm\(^3\)? (one-sample t-test)
Ex 2. Is the proportion of healthy adults with a right hippocampal volume > 3cm\(^3\) equal to 50%? (one-sample proportion test or z-test)
Ex 3. Do healthy men and women have the same mean left hippocampal volume? (two-sample t-test)
Ex 4. Is the left hippocampal volume equal to the right hippocampal volume in humans? (paired t-test)
Ex 5. Is having a large hippocampus (> 7 cm\(^3\)) associated with gender? (chi-squared test)
Ex 6. Is hippocampal volume correlated with age in healthy adults? (Pearson’s correlation or linear regression)
Ex 7. Does left hippocampal volume differ across diagnostic groups? (one-way ANOVA or linear regression)