Day125 - STAT Review: Revisiting Mathematics Theories (1)
Mathematical Principles: Hypothesis Testing, Paired Samples, Independent Samples, and additional concepts.
Hypothesis Test
- Confidence Intervals: Unknown Mean, Unknown Variance
- We often do not know what the variance $\sigma^2$ and the mean $\mu$ is.
- However, we can estimate $\sigma^2$ with the sample variance $s^2$.
- But we must be a bit more careful here because the sampling distribution for $\bar{X}$ is more variable, and the values $s^2$ are likely to differ from sample to sample.
- We can use the Student’s t distribution.
- Student t Distribution
- Recall that, assuming $n$ is sufficiently large, then $Z = \frac{\bar{X}-\mu}{\sigma / \sqrt{n}} \approx N(0,1)$
- If we replace $\sigma$ with s, we get $t = \frac{\bar{X}-\mu}{s/ \sqrt{n}}$, which is not a standard normal.
- Instead, t has a Student’s distribution with $n-1$ degrees of freedom, denoted $t_{n-1}$
- Student’s t Distribution: Properties
- Intuition: Student’s t distribution is similar to a standard normal distribution but has thicker tails (extreme values more likely).
- The t distribution is unimodal and symmetric about 0.
- The shape of the t distribution reflects the extra variability introduced by estimating the variance.
- The degrees of freedom (
df) measure the amount of information in the data that can be used to estimate $\sigma^2$- Because we lose one degree of freedom in estimating the mean (to estimate variance), we are left with $n-1$ df to estimate $\sigma^2$.
- Because we lose one degree of freedom in estimating the mean (to estimate variance), we are left with $n-1$ df to estimate $\sigma^2$.
- Student’s t Distribution: Degrees of Freedom
- For each possible value of the degrees of freedom, there is a different t-distribution
- When the degrees of freedom are low, the distribution is more spread out with heavier tails (a worse estimate means more variability)
- As the degrees of freedom approach infinity, the t distribution approaches the normal distribution.
- Intuition: if $n$ is very large, our estimate of $s^2$ is essentially the same as knowing $\sigma^2$
- Test Statistic
- Once we formulate our hypotheses, we need to draw a random sample of size $n$ from the population of interest.
- Calculate the sample statistic and compare it to the population parameter.
- Use test statistics to determine the probability of seeing a sample as extreme or more extreme than the one observed, given that the null hypothesis is true.
- Relies on the sampling distribution of the test statistics.
- p-value
- p-value is a statistical measurement used to validate a hypothesis against observed data.
- A p-value measures the probability of obtaining the observed results, assuming the null hypothesis is true.
- The lower the p-value, the greater the statistical significance of the observed difference.
- A p-value of 0.05 or lower is generally considered statistically significant.
- P-value can serve as an alternative to—or in addition to—preselected confidence levels for hypothesis testing.
- Significance
- We fail to reject the null hypothesis if it seems reasonable (i.e., not extremely unlikely) that the sample came from a population centered at the hypothesized mean.
- P-value calculates a probability to determine how unlikely it is to see your sample results if the null hypothesis is true.
- If that probability is less than some pre-specified significance level, $\alpha$, then reject the null hypothesis
- Significance level alpha
- Choosing the significance level $alpha$ allows us to specify the “power.”
- If we want to be more conservative, we can choose $\alpha=0.01$
- t-tests
- When $\sigma^2$ is also unknown, we substitute the sample variance $s^2$ and use the t distribution instead of the normal distribution.
- We determine the probability of seeing a test statistic t as extreme or more extreme as the one observed via a t_test.
Hypothesis Testing with Two Samples
- Comparison of two means
- We used a hypothesis test to compare the unknown mean of a single population to some fixed, known value, $\mu_0$
- Often, we want to compare the means of two separate populations where both means are unknown.
- For example, is the average height of Americans equal to the average height of Canadians?
- In two cases, samples are paired or independent
- Paired weight before and after surgery for a group of men
- Independence: heights of the Americans vs. Canadians
- Paired Samples
- For each observation in the first group, there is a corresponding observation in the second group.
- Self-pairing: measurements are taken on a single subject at two distinct time points (before and after)
- Matched pairing: match two individuals with similar demographics/ characteristics and compare their differences in response
- Depending on the setting, make the pair as similar as possible regarding essential characteristics (e.g., age, gender, socioeconomic status, etc.).
- Use pairing to control for extraneous sources of variation that otherwise influence results.
- By measuring the sample, we remove natural biological variability between people.
- If we pair based on a particular characteristic (e.g., age), we do not have to worry about that characteristic (age) influencing the results.
- In general, paring makes comparisons more precise
-
Paired Samples Procedure
- The data is paired so that we can use the difference $d_i = b_i - a_i$ as the data
- Reduces to the one-sample problem: compare differences to 0
- Mean: $\bar{x_d} (unknown true $\mu_d$), Standard Deviation $s_d$ (unknown true $\sigma_d$)
- Standard Error : $\frac{s_d}{\sqrt{n}}$
-
Paired Samples: R Code
t.test(difference) t.test(before, after, paired=T) - Independent Samples
- Suppose we have measurements on two samples of subjects
- Heart rates for a group of people who have been sitting
- Heart rates for a group of people who have been running
- The two underlying populations are independent and normally distributed
- The first population has mean $\mu_1$ and the second population has mean $\mu_2$
- Test whether the two populations are identical
- $H_0$ : $\mu_1 = \mu_2 $ vs. $H_1$: $\mu_1 =\mu_2$
- Suppose we have measurements on two samples of subjects
Leave a comment