Data Analysis for Graduate Research - Inference between two means

Model

Response data from observation belonging to group
The random error terms are independently and identically distributed (iid) as normal with mean zero and common variance.

2 sample T-test for difference in means between two independent groups

Parameter:
Estimate:
Assumptions:
- Group 1 & 2 are mutually exclusive and independent
- Difference is normally distributed
- Variance within each group are approximately the same ()

: There is no difference in the averages between groups.

: There is a difference in the averages between groups.

Example: BMI vs smoking

We would like to know, is there convincing evidence that the average BMI differs between those who have ever smoked a cigarette in their life compared to those who have never smoked?

Nitty gritty detail

For the purposes of learning, you will be writing out each step in the analysis in depth. As you begin to master these analyses, it is natural to slowly start to blend and some steps. However it is important for you to have a baseline reference.

1. Identify response and explanatory variables

Ever smoker = binary explanatory variable (variable eversmoke_c)
BMI = quantitative response variable (variable BMI)

2. Visualize and summarise

Show the code

plot.bmi.smoke <- addhealth %>% select(eversmoke_c, BMI) %>% na.omit()

plot.bmi.smoke %>% 
  ggviolin(x="eversmoke_c",
    y="BMI",
    color="eversmoke_c", 
    add = c("mean", "boxplot")) + 
  color_palette(palette = "jco") + xlab("Smoking Status")

Show the code

plot.bmi.smoke %>% 
  tbl_summary(
    by="eversmoke_c",
    digits = all_continuous() ~ 1,     
    statistic = list(
      all_continuous() ~ "{mean} ({sd})"
    ))

Characteristic	Never Smoked, N = 1,750¹	Smoked at least once, N = 3,276¹
BMI	29.7 (7.8)	28.8 (7.3)
¹ Mean (SD)

Smokers have on average BMI of 28.8, smaller than the average BMI of non-smokers at 29.7. Non-smokers have more variation in their weights (7.8 vs 7.3lbs), but the distributions both look normal, if slightly skewed right.

3. Write the null and research hypothesis in words and symbols.

Let be the average BMI for smokers, and be the average BMI for non-smokers

There is no difference in the average BMI between smokers and non-smokers.

There is a difference in the average BMI between smokers and non-smokers.

4. State and justify the analysis model. Check assumptions.

We are comparing the means between two independent samples. A Two-Sample T-Test for a difference in means will be conducted.
The assumptions that the groups are independent is upheld because each individual can only be either a smoker or non smoker.
The difference in sample means is normally distributed - this is a valid assumption due to the large sample size and that differences typically are normally distributed.
The observations are independent, this was a random sample
The variances are roughly equal (67/44 = 1.5 is smaller than 2).

5. Conduct the test and make a decision about the plausibility of the alternative hypothesis.

Show the code

t.test(BMI ~ eversmoke_c, data=addhealth)


    Welch Two Sample t-test

data:  BMI by eversmoke_c
t = 3.6937, df = 3395.3, p-value = 0.0002245
alternative hypothesis: true difference in means between group Never Smoked and group Smoked at least once is not equal to 0
95 percent confidence interval:
 0.3906204 1.2744780
sample estimates:
        mean in group Never Smoked mean in group Smoked at least once 
                          29.67977                           28.84722

There is strong evidence in favor of the alternative hypothesis. The interval for the differences (0.4, 1.3) does not contain zero and the p-value = .0002.

6. Write a conclusion in context of the problem. Include the point estimates, confidence interval for the difference and p-value.

On average, non-smokers have a significantly higher 0.82 (0.39, 1.27) BMI compared to smokers (p=.0002).

Inference between two means

Quantitative Outcome ~ Binary Covariate

Model

2 sample T-test for difference in means between two independent groups

Example: BMI vs smoking

1. Identify response and explanatory variables

2. Visualize and summarise

3. Write the null and research hypothesis in words and symbols.

4. State and justify the analysis model. Check assumptions.

5. Conduct the test and make a decision about the plausibility of the alternative hypothesis.

6. Write a conclusion in context of the problem. Include the point estimates, confidence interval for the difference and p-value.

Assumptions

But we could be wrong

But we could be wrong

Type I and Type II Error