Independent Samples T-Test

• An introduction to the t-test explains how to compare two groups using the independent samples t-test.
• The t-test is used to compare groups of people, like PhD vs. undergrad students' anxiety levels, men vs. women's drinking habits, and school district student performance.
• A random sample is split into two groups, G1 and G2, with a normally distributed numeric variable X measured, to determine if group membership is associated with different X values.
• The scientific question framed is whether the mean value of X differs between groups G1 and G2.
• µG1 represents the population-level mean of X for G1, and µG2 represents the same for G2.
• The core question is whether µG1 equals µG2.

Statistical Hypotheses

• The null hypothesis (H₀) typically states that µG1 equals µG2.
• The alternate hypothesis (HA) states that µG1 does not equal to µG2.
• µG1 = µG2 is equivalent to µG1 - µG2 = 0. Therefore, the hypotheses can be written as H₀: µG1 - µG2 = 0 and HA: µG1 - µG2 ≠ 0.
• The independent sample t-test assesses the probability of the observed data, assuming H₀ is true.

Logic of the t-test

• A statistical test is run assuming that the null hypothesis is true.
• If µG1 - µG2 = 0 at the population-level, sampling from G1 and G2 should result in a difference in group means of X close to 0.
• While population-level means are denoted by µG1 and µG2 , sample group mean values are represented by XG1 and XG2.
• Sampling never results in a perfect representation of the populations, so the difference in sample group means (XG1 - XG2) will likely not equal 0.
• If the null is true, the difference XG1 – XG2 should be close to 0, positive or negative.
• If the null hypothesis is true, values of XG1 - XG2 farther from 0 are less likely, and negative and positive values are equally likely, similar to a normal distribution.
• 0 is the most likely value in test
• Values closer to 0 are more likely than values farther from 0
• Positive and negative values appear equally as likely to occur (i.e. symmetry)

Student's t-Distribution

• A normal distribution is defined by a mean value μ and a standard deviation σ.
• It is common not to know the population-level standard deviation of X, especially in epidemiologic research.
• The t-distribution is a variation of the standard normal (Z) distribution, used when the population standard deviation is unknown.
• Any normal distribution (N(μ, σ)) can be transformed to the Z-distribution (N(0, 1)).
• The t-distribution standardizes the distribution
• The t-distribution is more conservative than the Z-distribution, assuming wider variability in observations.
• Less information leads to less certainty that observations are near the mean.
• The t-distribution is defined as a function of the degrees of freedom available to measure the variability in the data.

Degrees of freedom

• Degrees of freedom: number of parameters that can "vary freely" given an assumed outcome.
• With 100 participants and a known mean age of 60, there are numerous age possibilities that average to 60.
• If the exact age of 99 individuals out of 100 is known, the final person's age cannot "vary freely".
• Only one value can allow the average to be 60. Therefore, calculating a mean "spends" one degree of freedom.
• With n observations, calculating sample-mean x spends one degree of freedom, leaving n - 1 degrees of freedom to calculate standard deviation s.
• Fewer observations mean less information to estimate the variation of observed variable X.
• The t-distribution captures uncertainty in standard deviation measurement from a small sample.
• Fewer degrees of freedom mean less certainty that measured standard deviations represents population-level standard deviation σ.
• The t-distribution is "shorter" and "wider" than the normal distribution to capture this. Values farther from 0 are more likely than under the Z-distribution.
• As more data is collected, the t-distribution approaches the shape of the Z-distribution
• As the number of observations n increases, the t-distribution resembles more like the normal distribution.
• When n ≥ 30, the t-distribution is assumed to be about same as normal distribution.

Calculating the t-statistic

• Our null hypothesis is that µG1 - µG2 = 0
• If our null hypothesis is true, XG1 - XG2 looks like a normally distributed variable.
• We don't know the population-level standard deviation, so assume that XG1 - XG2 follows a t-distribution, with wider tails.
• Letting nG1 represent number of people in G1 if nG2 represents number of people in G2, then use a t-distribution with (nG1 - 1) + (nG2 - 1) degrees of freedom. That is because we have (nG1 - 1) degrees of freedom to calculate the variability of X among G1, and (nG2 – 1) to calculate that among G2.

Mapping the Signal onto the t-Distribution

• The aim is to determine probability of data under the null hypothesis = µG1 = µG2
• Our signal is the difference in mean value of X across groups, or: XG1 – XG2.
• The likely value of the signal is 0, which means distribution is centered around 0 (similar to Z-distribution).
• Must scale signal by noise in data, standardizing the signal to fit the appropriate t-distribution, which is a Z-distribution variation with standard deviation= 1.
• Standardize signal by dividing it by the standard error of the mean of observed values of X.
• Standard error is an estimation of the population-level standard deviation, getting more precise as sample size grows.
• The standard error indicates the population-level standard deviation becomes more precise as the sample size increases.

Typical standard error equation

• SE = s / √n
• s is the sample standard deviation.
• Due to us comparing two groups a slight variation of equation is used; SE = s / √(nG1 + nG2)
• The equation will derive from the measured variance s², representing average distance of each n observations of X from the mean value x.
• An assumption of the independent samples t-test is that the population-level variance of variable X is similar for both groups
• Must calculate the pooled variance of X within both G1 and G2 like so; s² = ((nG1 - 1) * sG1 + (nG2 – 1) * sG2) / (ng1 + ng2 – 2)
• The sample variance of X for G1 = sG1 and sample variance of X for G2 = sG2
• The square root formula is taken to identify the standard deviation; S = √s² = √((nG1 – 1) * s1 + (nG2 – 1) * sG2) / (ng1 + ng2 – 2))
• This test has: ng 1 + ng2 – 2 degrees of freedom to estimate the variability in our data. The equation above displays prominent number. Standard error is then calculated.

Calculating Test Statistic

• t= (XG1 - XG2) / SE formula for standardized value and this equation will formalize standardized value for test.
• Standard error formula ; t = (XG1 - XG2) / (s * √(1/NG1 + 1/NG2)
• By calculating t, signal is taken (XG1 - XG2) and standardized it to a t-distribution.
• There are NG1 + NG2 2 degrees of freedom!
• The value is mapped onto this distribution for area calculations.
• Comparing the calculated t to a t-distribution with NG1 + NG2 2 degrees of freedom will test the likelihood of observing data or more extreme assuming null is true.
• t = −2.36 calculated on a t-distribution with 100 + 100 - 2 = 198 degrees of freedom.

Two-Tailed Versus One-Tailed Test

• The previous image can be referred to as a two-tailed t-test because extreme values are checked on both tails of distribution.
• Generally we do a two-tailed test when want to know whether the mean values are different
• There are cases where it is assumed that the effect only occurs in one direction, where the reduction in treatment group is greater than other control group.
• Alternate hypothesis would instead indicate the increase that μτ > μc,
• If we ran a t-test to test this and got t = 2.3, need to calculate area under the curve for the positive tail.

Three Variations of the t-test

Independent Samples t-test

• Compares the mean value of a random normally distributed variable X between two groups G1 and G2
• The formula is t = (XG1 - XG2) / s * √(1/NG1 + 1/NG2) , comparing value t to t-distribution with NG1 + NG2 2 degrees of freedom. A p-value is found. Taking area under curve finds all values that are more extreme than observed test t statistic.
• Must write hypotheses. H₀: µG1 = µG2 and HA: µG1 ≠ µG2
• Significant finding supplies with evidence that null hypothesis is incorrect because observed data is unlikely.

One Sample t-test

• It is done to compare the mean value of a normally distributed variable X of a group G to a specific value.
• To test method, decide to see if the mean weight of 100 bags of flour weighs 1 pound. Then our hypotheses will say that H0: μ = 1 and H₁: μ ≠ 1
• Average weight of bag of flower = u. To run the 1-sample t-test, we calculate the mean of 100 bags = x and the standard deviation of observations = s. We then calculate t as follows:
• t= (x-1) / (s*√(1/n)) Then calculate p by comparing t to a t-distribution with n 1 degrees of freedom.

Paired Samples t-test

• Takes same measurement from the sample at two separate time points the assess mean value
• Assesses if that mean value changed or remained same. Intervention helps participants performance.
• For each participant measure a variable at two time points. From time 1 x₁ and time 2 x₂, want to measure the difference in their score.
• Define difference as d = X1 X2
• Calculate mean value of d for all participant ā and the standard deviation of d, są. Our null and alternate hypotheses are identified as:
• Ho: d = 0 and H₁: d ≠ 0
• Then calculate t test statistic as; t = d / (Sd*√(1/n))
• Comparing t to a t-distribution with n 1 degrees of freedom identifies P-value

Assumptions for running the Independent Samples T-test

• To run test need to ensure that our data is appropriate to use, and if any assumptions are made
• Let variable = X

Variable of Interest Needs Measured on an Ordinal or Continuous Scale

• Must be measured on an ordinal or continuous scale.
• If it is a variable of interest then it cannot run t test
• Use Summary() Function and look at visual dataset.

Data needs to be Drawn from a Random Sample.

• The effectiveness of the t-test depends on giving an effective random sample.
• Sampling Risk discussed in research methods cause bias
• If there is bias, the the t-test could expose this bias.

Groups need to be independent

• Independent Samples t-test: two groups need to be independent from each other
• Need to represent distinct populations for both.

Normality of Observations

• For each group being study the mean value of X has to be study.
• Given all the assumptions, the larger our sample size is, the weaker assumption the t-test becomes
• If the method is robust then certain violations still result in the data being underlying.

Homogeneity of Variance

• Assumed that independent samples t-tests are variance, along with assuming standard value
• Levene Test identifies the variance of X for the sample group.
• Use leveneTest() in“car” package, data must be structured by a data.frame.
• When we created our dataframe and bound our columns using coding lines

Code for Running A t-test In R

• T-test in R needs two code vectors
• Run test from t.test() function. Has been learned after that so test helps assist

This quiz covers the Independent Samples T-Test. It reviews the purpose, null hypothesis, assumptions, and interpretation of results. Questions cover understanding the test's underlying principles and statistical conclusions.