Independent Samples T-Test

• A common objective is to compare groups of people
• The t-test is one such test, for when the data calls for it

Comparing Two Groups

• Split a random sample into two groups, G₁ and G₂
• Measure a numeric variable X, which we assume is normally distributed
• Determine if group membership (G₁ U G₂) is associated with different values of X
• Frame this as a scientific question: "Is the mean value of X different between groups G₁ and G₂?"
• Define µG₁ as the population-level mean of X for G₁, and µG₂ as that for G₂
• Determine if µG₁ = µG₂

Statistical Hypotheses of the Independent Samples T-Test

• The null and alternate hypothesis of the t-test include:
• H₀: µG₁ = µG₂
• HA: µG₁ ≠ µG₂
• µG₁ = µG₂ is the same as µG₁ - µG₂ = 0 Therefore, the null and alternate hypothesis can be expressed as:
• H₀: µG₁ - µG₁ = 0
• HA: µG₁ - µG₁ ≠ 0
• The independent sample t-test will estimate the probability of observed data assuming that H₀ is true

Logic of the t-test

• When a statistical test is run, the null hypothesis is assumed to be true
• If the null is true and that µG₁ - µG₂ = 0 at the population-level, then if a set of people from G₁ and G₂ are sampled and the difference in their group means of X is looked at, 0 is the most likely value
• µG₁ and µG₂ represent the population-level mean, and X̄G₁ and X̄G₂ to represent the mean values of sample groups
• Sampling never results in a perfect representation of the populations, so the variance in sample group means (X̄G₁ - X̄G₂) won't be 0
• If the null is true then it would make sense if X̄G₁ – X̄G₂ was close to 0 (whether positive or negative)
• If the null hypothesis is true, then we would assume values of X̄G₁ - X̄G₂ further from 0 would be less likely than values closer to 0
• Negative and positive values are equally likely
• The normal distribution includes;
• 0 is the most likely value
• Values closer to 0 are more likely than values further from 0
• Positive and negative values appear equally as likely to occur (i.e. symmetry)

Introducing the Student's t-Distribution

• A normal distribution is defined by a mean value u and a standard deviation σ
• It is unknown if the population-level standard deviation of X is known, even if a normally distributed variable X has been measured
• This is fairly common in epidemiologic research, because population-level data is unavailable about research groups
• The t-distribution is a variation of the standard normal distribution (Z-distribution), to be used when the standard deviation of the population is unknown
• Any normal distribution (N(μ, σ)) can be transformed to the Z-distribution (N(0, 1))
• The t-distribution can be understood as a standardized distribution
• The t-distribution is a conservative version of the Z-distribution, where a wider variability in observations is assumed
• When there is less information, there is less certainty that observations will be near the mean
• The t-distribution is defined as a function of the number of degrees of freedom available to measure the variability in data

Degrees of Freedom

• Degrees of freedom refer to the number of parameters that are able to "vary freely", given some assumed outcome
• Given 100 participants with a mean average og 60 years old, there are possibilities for how 100 participants age could average out to 60
• People's ages can "vary freely” while still maintaining an average age of 60 years old
• If the exact age of 99 of the 100 individuals is known, with an average age of 60 years, the age of the final person cannot “vary freely”
• There is only one value that could take the first 99 values and get the average to 60 years of age
• If the average age of the first 99 people is 60 years of age, then the age of the final person must be 60 years of age.

Observations and Measurement

• A normal distribution is defined by a mean value u and a standard deviation σ.
• If there are n observations and sample-mean x̄ and standard deviation s are measured, one degree of freedom must be spent to calculate x̄.
• This means there are n - 1 degrees of freedom to calculate s.
• The fewer observations there are (i.e., the smaller that n is), the less information there is to estimate the variation of our observed variable X.
• As such, the t-distribution is intended to capture uncertainty in the measurement of the standard deviation from a small sample.
• The fewer degrees of freedom (i.e., the smaller our sample), the less certain that our measured standard deviation s represents our population-level standard deviation
• The t-distribution is “shorter” and “wider” than the normal distribution.
• Under the t-distribution, values farther from 0 are more likely than under the Z-distribution.
• As we collect more data (i.e., as n gets larger), the t-distribution's shapes approaches that of the Z-distribution.
• As the number of observations n increases, the t-distribution begins to look a lot like the normal distribution.
• If n ≥ 30 then assume that the t-distribution is the same as the normal distribution.

Null Hypothesis

• The null hypothesis is that µG₁ - µG₂ = 0
• X̄G₁ - X̄G₂ appears to behave like a normally distributed variable
• Since we do not know the population-level standard deviation, assume that X̄G₁ - X̄G₂ follows a t-distribution, which has wider tails
• If nG₁ represents the number of people in G₁ and if nG₂ represent the number of people in G₂, then we use a t-distribution with (nG₁ – 1) + (nG₂ − 1) = nG₁ + nG₂ – 2 degrees of freedom
• This 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

• Use probability of data under the null hypothesis that µG₁ = µG₂
• Our signal is the difference in mean value of X across groups, or: X̄G₁ – X̄G₂
• The most likely value of signal, assuming the null is true, is 0, so its corresponding distribution is centered around 0 (like the Z-distribution)
• Scale signal by the noise in the data, standardize the signal to correspond to the appropriate t-distribution, which is a variation of the Z-distribution whose standard deviation is 1

Calculating the Standard Error of the Mean

• To standardize signal, divide it by the standard error of the mean of the observed values of X
• The standard error is an estimation of population-level standard deviation, which gets more precise (or smaller) as the sample size (n) gets larger
• The equation for the standard error is: SE = s/√n
• When comparing two groups, use a variation on this equation: SE = s/√nG₁ + nG₂
• s is the sample standard deviation and is derived from the measured variance s², which represents the average distance of each n observations of X from the mean value x
• s² = 1/n-1 * Σ(xi – x̄)
• Comparing two groups G₁ and G₂: s² = ((nG₁ - 1) * s²G₁ + (nG₂ – 1) * s²G₂) / nG₁ + nG₂ – 2
• Where s²G₁ is the sample variance of X for G₁ and s²G₂ is that for G₂
• To get the sample standard deviation, take the square root: s = √s² = √((nG₁ – 1) * s²G₁ + (nG₂ – 1) * s²G₂) / nG₁ + nG₂ – 2
• We have nG₁ + nG₂ – 2 degrees of the measure the variability in our data

Testing the Statistic t

• To test standardize X̄G₁ - X̄G₂:
• t = (X̄G₁ - X̄G₂) / SE Often calculated by using standard error in the equation:
• t = (X̄G₁ - X̄G₂) / s * √(1/nG₁) + (1/nG₂)
• This value can be mapped onto this distribution and make area under the curve calculations

T Distribution

• The t-statistic with nG₁ + nG₂ - 2 degrees of freedom!
• Used to map calculated value onto the distribution and make area
• Given a sample of 100 people in G₁ and 100 people G₂: x̄₁ = 21 x̄₂ = 22.
• Pooled standard deviation of 3

Two-Tailed T-Test

• A two-tailed t-test looks at extreme values in both tails of the distribution
• Generally do a two-tailed test when we want to know if the mean values are different

One Tailed T-Test

• Considers that an affect can only occur in one direction
• Instead of the alternate hypothesis being that µT ≠ µC, then μT > μC

T-Test Variations

• Independent Samples t-test
• Determine mean value of a random normally distributed variable X : t = (X̄G₁ - X̄G₂) / s√(1/nG₁) + (1/nG₂)
• Compare t to t-distribution with nG1 + nG2 2 degrees of freedom.
• P-value is computed by taking area under the curve for all values more extreme than the observed test statistic t. In this case, our hypotheses are : H₀: µG1 = µG2 and HA: µG1 ≠ µG2 Significant finding indicates our data is unlikely, if H₀ is assumed. This supplies evidence that H₀ is incorrect.

One Sample T Test

• Used to compare mean value of variable X of group G to a specific value.
• H₀*: µ = 1
• H₁*: u ≠ 1

Calculate mean and the standard deviation of 100 bags of flour; x = 1 and calculate the standard deviation of observations s and we calculate t s as:

• t* = (𝑥 - 1) / (s / √𝘯 )

Then calculate compute the p-value by comparing t to a t-distribution with n - 1 degrees.

Paired Sample T Test

• Done when we take the measurement from the sample at two separate time points and we wish to assess: the mean value has changed or remained
• This assesses if an intervention has changed participants performance on a knowledge-based task.
• For each participant : score from time 1, x1, and time 2 x2 is measured
• Subtract the distance between the data: d = Χ1- Χ2
• H0*: d = 0
• HA*: d ≠ 0

Calculate the t test statistic :

𝑡 = 𝑑 /( S𝑑∗  / 𝑛)

Then calculate the p-value by comparing t to a t-distribution with n - 1 degrees of freedom.

Assumptions for Running Independent Samples

• Test data is appropriate for doing
• Must know what assumptions must be met by our data
• Need to be able to check if assumptions are met

These steps have one variable: X

1. Our variable of interest: X must be Measured on an Ordinal or Continuous Scale

• X: continuous scale.
• Cannot run t-test when variable is categorical
• Summary()* function is a good place to start

2. Data Must Be Drawn From Random Sample:

• Effective t-test depends on administering an effective random sampling

3. Groups Must Be Independent:

• Two groups must be independent of each other
• Two groups must represent distinct populations.
• That would indicate certain individuals to represent both groups.

4. Normality of observations of X:

• Overall sample size: the weaker”, this assumption the sample becomes. Test has proven robust.

5. Homogeneity Of Variance:

• t-test assumes: variance of the two groups is the same.
• that value are normally distributed
• Levene Test: for groups’ same.

Running a t-Test in R

• Given vector & variables two created measured.

1. Contains all X values first. 𝑁𝑋  G.

2. Contains all X values second. 𝑁𝑋  G2.

• To test() function* as functions those them. Should generated frame pressure blood, then run levene test. All observations pressure blood: vector with shall such As. Test() run to them supplied, those and women men of.

This content covers independent samples t-tests. It discusses the interpretation of rejecting the null hypothesis, the effect of sample size, and interpreting p-values. It also describes a scenario for its appropriate use.