Séance 8 : Analyse de la Variance

Session 8: Analysis of Variance

• Determining if the means of two groups are significantly different is done using a t-test.
• Determining if the means of more than two groups are significantly different is done using an analysis of variance (ANOVA).

T-test and Analysis of Variance

• Two methods for evaluating the statistical significance of difference(s) between sample means.
• Does the difference(s) exist in the population as well?
• Is the difference(s) improbable enough, considering the null hypothesis?

Multiple T-tests

• Using multiple t-tests for comparisons between groups increases the probability of a Type I error (incorrectly rejecting a true null hypothesis).
• ANOVA is a better approach when comparing multiple groups.

Analysis of Variance (ANOVA)

• A single test that evaluates the significance of differences between several samples' means.
• The risk of a Type I error remains constant at 5%.

Political Information Level (0-4)

• Data (2015 Vote) from a survey.
• This analysis evaluates the likelihood that all the groups' means in the sample came from a population where the groups' means were the same.
• This is based on the "null hypothesis."

Variance

• Two primary categories:
  • Intergroup variance: Represents the variance between the group means.
  • Intragroup variance: Represents the variance within the groups.
• Intergroup variance measures the variance between the different group means and between the group means and the overall mean.

ANOVA Table

• Indicates whether the significance is less than 0.05.
• If yes, the probability of finding such a relationship, assuming the means are the same in the population, is low (significance is likely).
• The null hypothesis is rejected.
• The sample means are likely different.
• If no, the probability of finding such a link, assuming the means are the same in the population, is not low (not enough significance).
• The null hypothesis is not rejected.
• The sample means are not likely different.

Bivariate Analysis (Interval/Ratio Variables)

• Used to determine if one variable could explain the other.
• Three tools for analyzing the relationship:
  • Scatter plot: A graphical tool to illustrate the relationship between two interval/ratio variables in terms of direction, strength, and form.
  • Correlation coefficient: A single number that summarizes the direction and strength of a relationship between two variables, but does not show the shape.
  • Regression analysis: Used in several areas in order to predict values of one variable from another.

Correlation Coefficient (r)

• A measure of the linear relationship between two interval/ratio variables.
• It ranges from -1 (perfect negative relationship) to +1 (perfect positive relationship), with 0 indicating no linear relationship.
• The value summarizes the direction (positive or negative) and strength (weak or strong) of the linear relationship between variables.

Significance Test of Correlation Coefficient (Test F)

• The level of statistical significance of the correlation is determined by a test called F.
• A high F-stat value indicates a significant correlation (usually a cutoff of 3.84).

Linear Regression Equation

• A formula to estimate (predict) values of one variable (dependent) based on the known values of another variable (independent). It helps determine how much one variable affects the other.
• Uses a constant (to predict in case of the dependent variable has the value 0) and a correlation coefficient (indicates whether the relationship between two variables is positive or negative) or slope.

Statistical Significance (T-test)

• A test that checks if the relationship between the two variables is statistically significant (i.e., not due to chance).
• A threshold value of 1.96 or higher for the absolute value of a t-value suggests statistical significance.