Why statistical hypothesis testing is important?

Hypothesis testing provides a reliable framework for making any data-driven decisions for your population of interest.

Step 1

Establish null and alternative hypotheses

$$H_1:\mu <12.5kg$$

Step 1

Establish null and alternative hypotheses

$$H_0:\mu \ge 12.5kg$$

$$H_1:\mu <12.5kg$$ $\mu$ - population mean

Step 2: Gather sample data

gas

## # A tibble: 35 × 1
##    weight
##     <dbl>
##  1   9.44
##  2   9.77
##  3  11.6 
##  4  10.1 
##  5  10.1 
##  6  11.7 
##  7  10.5 
##  8   8.73
##  9   9.31
## 10   9.55
## # … with 25 more rows

Step 3: Visualize data

library(ggplot2)
ggplot(gas, aes(x=weight)) +  geom_boxplot(alpha=0.5)

ggplot(gas, aes(x=weight)) +  geom_boxplot(alpha=0.5) +
  theme( # remove axis text and ticks
    axis.text.y = element_blank(), axis.ticks = element_blank()) + labs(x="weight")

Step 4: Determine the appropriate statistical test

$$n>30$$

Central Limit Theorem

Distribution of data unknown/ If sample size is small, perform normality test.

Method 1

H0: The data are normally distributed.

H1: The data are not normally distributed.

Method 2

Let $X$ be the weight of a randomly selected cylinder

H0: $X$ is normally distributed

H1: $X$ is not normally distributed

Normal-probability plot

ggplot(gas, aes(sample = weight)) +
    stat_qq() + stat_qq_line(col="red") + coord_equal() + ylab("Theoretical Quantiles") + xlab("Sample Quantiles")

How to perform normality test in R?

#perform shapiro-wilk test
shapiro.test(gas$weight)

## 
##     Shapiro-Wilk normality test
## 
## data:  gas$weight
## W = 0.98495, p-value = 0.9027

Step 5: Test

"The R package statsr provides functions and datasets to support the Coursera Statistics with R Specialization videos and open access book An Introduction to Bayesian Thinking for learning Bayesian and frequentist statistics using R"

source: https://www.rdocumentation.org/packages/statsr/versions/0.3.0

statsr package installation

library(devtools)
devtools::install_github("statswithr/statsr",
                  dependencies=TRUE,
                      upgrade_dependencies = TRUE)

Method 1: Using statsr package

library(statsr)
inference(weight, data=gas, statistic="mean", type="ht", null=12.5,
          alternative ="less", method="theoretical")

## Single numerical variable
## n = 35, y-bar = 10.0375, s = 0.9463
## H0: mu = 12.5
## HA: mu < 12.5
## t = -15.3956, df = 34
## p_value = < 0.0001

## Single numerical variable
## n = 35, y-bar = 10.0375, s = 0.9463
## H0: mu = 12.5
## HA: mu < 12.5
## t = -15.3956, df = 34
## p_value = < 0.0001

Confidence intervals with statsr

inference(weight, data=gas, statistic="mean", type="ci",  method="theoretical")

## Single numerical variable
## n = 35, y-bar = 10.0375, s = 0.9463
## 95% CI: (9.7125 , 10.3626)

Method 2: Using stats package

t.test(gas$weight, mu=12.5, alternative = "less")

## 
##     One Sample t-test
## 
## data:  gas$weight
## t = -15.396, df = 34, p-value < 2.2e-16
## alternative hypothesis: true mean is less than 12.5
## 95 percent confidence interval:
##      -Inf 10.30798
## sample estimates:
## mean of x 
##  10.03752

Confidence intervals with stats

t.test(gas$weight, mu=12.5, alternative = "two.sided")

## 
##     One Sample t-test
## 
## data:  gas$weight
## t = -15.396, df = 34, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 12.5
## 95 percent confidence interval:
##   9.712466 10.362569
## sample estimates:
## mean of x 
##  10.03752

Example 2

A chemist wants to measure the bias in a pH meter. She uses the meter to measure the pH in 14 neutral substances (pH=7) and obtains the data below.

ph <- c( 7.01, 7.04, 6.97, 7.00, 6.99, 6.97, 7.04, 7.04, 7.01, 7.00, 6.99, 7.04, 7.07, 6.97)

Is there sufficient evidence to support the claim that the pH meter is not correctly calibrated at the α = 0.05 level of significance?

ph.df <- data.frame(pH=ph)
ggplot(ph.df, aes(y=pH, x="")) +
geom_boxplot(outlier.shape = NA, fill="forestgreen") +
geom_jitter(alpha=0.5) +
labs(x = "")

In this case, we have only sixteen observations, meaning that the Central Limit Theorem does not apply. With a small sample, we should only use the t-test if we can reasonably assume that the population data is normally distributed. Hence, we must first verify that pH is normally distributed.

ggplot(ph.df,
aes(sample=pH))+
stat_qq() + stat_qq_line()+labs(x="Theoretical Quantiles", y="Sample Quantiles") +
  theme(aspect.ratio=1)

shapiro.test(ph.df$pH)

## 
##     Shapiro-Wilk normality test
## 
## data:  ph.df$pH
## W = 0.91603, p-value = 0.1927

Hypothesis to be tested

$$H_0:\mu =7$$

$$H_1:\mu \ne 7$$

$\mu$ - Population mean pH value (in neutral substances).

Hypothesis test

inference(y=pH, data=ph.df, statistic="mean", type="ht", null=12.5,
          alternative ="twosided", method="theoretical")

## Single numerical variable
## n = 14, y-bar = 7.01, s = 0.0316
## H0: mu = 12.5
## HA: mu != 12.5
## t = -649.5856, df = 13
## p_value = < 0.0001

Hypothesis test

## Single numerical variable
## n = 14, y-bar = 7.01, s = 0.0316
## H0: mu = 12.5
## HA: mu != 12.5
## t = -649.5856, df = 13
## p_value = < 0.0001

Confidence interval

inference(y=pH, data=ph.df, statistic="mean", type="ci",  method="theoretical")

## Single numerical variable
## n = 14, y-bar = 7.01, s = 0.0316
## 95% CI: (6.9917 , 7.0283)

Confidence interval

## Single numerical variable
## n = 14, y-bar = 7.01, s = 0.0316
## 95% CI: (6.9917 , 7.0283)

Two-sample: Paired t-test

A dietician hopes to reduce a person’s cholesterol level by using a special diet supplemented with a combination of vitamin pills. Twenty (20) subjects were pre-tested and then placed on diet for two weeks. Their cholesterol levels were checked after the two week period. The results are shown below. Cholesterol levels are measured in milligrams per decilitre.

cont.

i) Test the claim that the Cholesterol level before the special diet is greater than the Cholesterol level after the special diet at α = 0.01 level of significance.

ii) Construct 99% confidence interval for the difference in mean cholesterol levels. Assume that the cholesterol levels are normally distributed both before and after.

id <- 1:20
before <- c(210, 235, 208, 190, 172, 244, 211, 235, 210,
190, 175, 250, 200, 270, 222, 203, 209, 220, 250, 280)
after <- c(190, 170, 210, 188, 173, 195, 228, 200, 210, 184,
196, 208, 211, 212, 205, 221, 240, 250, 230, 220)
cholesterol_1 <- tibble(id=id, before=before, after=after)
head(cholesterol_1)

## # A tibble: 6 × 3
##      id before after
##   <int>  <dbl> <dbl>
## 1     1    210   190
## 2     2    235   170
## 3     3    208   210
## 4     4    190   188
## 5     5    172   173
## 6     6    244   195

library(tidyverse)
cholesterol_2 <- pivot_longer(cholesterol_1, before:after, "type", "value")
head(cholesterol_2)

## # A tibble: 6 × 3
##      id type   value
##   <int> <chr>  <dbl>
## 1     1 before   210
## 2     1 after    190
## 3     2 before   235
## 4     2 after    170
## 5     3 before   208
## 6     3 after    210

ggplot(data= cholesterol_2, aes(x=type, y=value)) +
geom_boxplot(outlier.shape = NA, aes(fill=type), alpha=0.5) +
geom_jitter(aes(fill=type))

cholesterol_2 %>%
group_by(type) %>%
summarize(mean = round(mean(value), 2),
sd = round(sd(value), 2))

## # A tibble: 2 × 3
##   type    mean    sd
##   <chr>  <dbl> <dbl>
## 1 after   207.  21.0
## 2 before  219.  29.3

ggplot(data = cholesterol_2, aes(sample = value)) +
stat_qq() +
stat_qq_line() +
facet_grid(. ~ type)

shapiro.test(cholesterol_1$before)

## 
##     Shapiro-Wilk normality test
## 
## data:  cholesterol_1$before
## W = 0.9647, p-value = 0.6414

shapiro.test(cholesterol_1$after)

## 
##     Shapiro-Wilk normality test
## 
## data:  cholesterol_1$after
## W = 0.98535, p-value = 0.9836

Method 1: stats package

t.test(before, after, data=cholesterol_1, "greater", paired=TRUE)

## 
##     Paired t-test
## 
## data:  before and after
## t = 1.7754, df = 19, p-value = 0.04593
## alternative hypothesis: true mean difference is greater than 0
## 95 percent confidence interval:
##  0.3167385       Inf
## sample estimates:
## mean difference 
##           12.15

Method 2: statsr package

diff <- cholesterol_1$before -cholesterol_1$after
diff_data <- data.frame(diff=diff)
ggplot(diff_data, aes(sample=diff))+
stat_qq() + stat_qq_line()+
labs(x="Theoretical Quantiles", y="Sample Quantiles") +  theme(aspect.ratio=1)

inference(y=diff,  data=diff_data,statistic="mean",  type="ht", method="theoretical", alternative ="greater", null=0L)

## Single numerical variable
## n = 20, y-bar = 12.15, s = 30.6049
## H0: mu = 0
## HA: mu > 0
## t = 1.7754, df = 19
## p_value = 0.0459

Method 2: stats

t.test(x = diff_data$diff, alternative = c("greater"), mu=0)

## 
##     One Sample t-test
## 
## data:  diff_data$diff
## t = 1.7754, df = 19, p-value = 0.04593
## alternative hypothesis: true mean is greater than 0
## 95 percent confidence interval:
##  0.3167385       Inf
## sample estimates:
## mean of x 
##     12.15

Two sample tests and ANOVA: switch to this