Independent Samples T-Test Explained

• The material is an introduction to the T-Test, a method for comparing two groups using independent samples.

Comparing Two Groups w/ Independent Samples T-test

• The t-test is a statistical test used to compare groups of people.
• A numeric variable X, assumed to be normally distributed, is measured.
• It assesses if group membership ( (G_1 \cup G_2) ) is associated with different values of X.
• The goal is to determine if ( \mu_{G1} = \mu_{G2} ), where ( \mu_{G1} ) and ( \mu_{G2} ) are the population-level means of X for groups ( G_1 ) and ( G_2 ) respectively.

Statistical Hypotheses of the Independent Samples T-Test

• Null hypothesis ( (H_0) ): ( \mu_{G1} = \mu_{G2} )
• Alternate hypothesis ( (H_A) ): ( \mu_{G1} \neq \mu_{G2} )
• The null and alternate hypotheses can also be expressed as:
  • ( H_0 ): ( \mu_{G1} - \mu_{G2} = 0 )
  • ( H_A ): ( \mu_{G1} - \mu_{G2} \neq 0 )
• The test assesses the likelihood of observed data assuming ( H_0 ) is true.

Logic of the t-test

• The null hypothesis is assumed to be true when running a statistical test.
• If ( \mu_{G1} - \mu_{G2} = 0 ) at the population level, the difference in group means of X should be 0.
• ( \overline{X}{G1} ) and ( \overline{X}{G2} ) represent the mean values of sample groups, contrasting with ( \mu_{G1} ) and ( \mu_{G2} ), which represent population-level means.
• The difference in sample group means ( (\overline{X}{G1} - \overline{X}{G2}) ) is unlikely to equal 0 due to sampling variability.
• If the null hypothesis is true, ( \overline{X}{G1} - \overline{X}{G2} ) should be close to 0, with negative and positive values being equally likely.

Introducing the Student's t-Distribution

• Normal distribution is defined by mean value ( \mu ) and standard deviation ( \sigma ).
• The t-distribution is a variation of the standard normal (Z) distribution, used when the population standard deviation is unknown.
• Any normal distribution ( N(\mu, \sigma) ) can be transformed to the Z-distribution ( N(0, 1) ).
• The t-distribution assumes wider variability in observations, making it a more conservative version of the Z-distribution.
• Observations are less certain to be near the mean when less information is known.

Degrees of freedom

• Degrees of freedom refer to the number of parameters able to vary freely given an assumed outcome.

• Calculating a mean "spends" one degree of freedom.

• Fewer observations (smaller n) lead to less information for estimating the variation of the variable X.

• The t-distribution is shorter and wider than the normal distribution, capturing uncertainty in standard deviation measurement from a small sample.

• Values farther from 0 are more likely under the t-distribution compared to the Z-distribution because less is known.

• As more data is collected, the t-distribution shape approaches that of the Z-distribution.

• The t-distribution is similar to the normal distribution but shorter and wider.

• In t-distribution, observing a value of 0 is less likely but observing extreme values is more likely than in a normal distribution.

• As the number of observations increases, the t-distribution resembles a normal distribution more closely.

• In practice, if ( n \geq 30 ), the t-distribution is assumed to be the same as the normal distribution .

Mapping the Signal onto the t-Distribution

• The goal is to assess how probable the data is given the null hypothesis ( \mu_{G1} = \mu_{G2} ).
• The signal is the difference in mean value of X across two groups ( (\overline{X}{G1} - \overline{X}{G2}) ).
• Standardize the signal to correspond to the appropriate t-distribution with a standard deviation of 1.

Calculating the Standard Error of the Mean

• Standardize the signal by dividing it by the standard error of the mean of observed X values.
• Standard error estimates the population-level standard deviation and becomes more precise with larger sample sizes.

Calculating Our Test Statistic t

• Formula for the t-statistic:
  • ( t = \frac{\overline{X}{G1} - \overline{X}{G2}}{SE} )
• The signal ( (\overline{X}{G1} - \overline{X}{G2}) ) has been taken and standardized it to a t-distribution with ( n_{G1} + n_{G2} - 2 ) degrees of freedom.

Two-Tailed Versus One-Tailed T-Test

• The two-tailed t-tests look at extreme values in both tails of the distribution.
• One-tailed tests are used when the effect can only occur in one direction.

Three Variations of the t-test

• Independent Samples t-test: Compares the mean value of a random normally distributed variable X between two groups ( G_1 ) and ( G_2 ).
  • The hypotheses are as follows:
    • (H_0): ( \mu_{G1} = \mu_{G2} )
    • (H_A): ( \mu_{G1} \neq \mu_{G2} )
• One Sample t-test: Compares the mean value of a normally distributed variable X of a group G to a specific value.
  • The hypotheses are as follows:
    • (H_0): ( \mu = 1 )
    • (H_A): ( \mu \neq 1 )
• Paired Samples t-test: Assesses if the mean value has changed or remained the same when the same measurement is taken from the same sample at two separate time points.
  • The hypotheses are as follows:
    • (H_0): ( \overline{d} = 0 )
    • (H_A): ( \overline{d} \neq 0 )

What Are the Assumptions We Make Prior to Running an Independent Samples T-test

• Each of the assumptions must be met by the data for running a t-test.
• The assumptions are:
  • Our variable of interest, X, must be measured on an ordinal or continuous scale.
  • Data must be drawn from a random sample.
  • Groups Must Be Independent
  • Normality of Observations of (X): The mean value of X for each group being studied should be normally distributed.
  • Homogeneity of Variance: The independent sample t-test assumes that the variance of the two groups is the same.

Understand the independent samples t-test, a statistical method used to compare the means of two independent groups. Learn about its applications and statistical hypotheses related to population means. Explore how it assesses the likelihood of observed data.