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Relationships and Differences Between Groups in Inferential Statistics -…
Relationships and Differences Between Groups in Inferential Statistics
Independent-Measures t test: A hypothesis test comparing the mean difference between two separate, unrelated groups.
Independent-Measures Design (Between Subjects): A design using completely separate groups of participants for each treatment condition.
Repeated-Measures Design (Within Subjects): A design in which the SAME group of participants provides data for all treatment conditions.
Null Hypothesis (H0): μ₁ − μ₂ = 0: States there is no mean difference between the two populations being compared.
Alternative Hypothesis (H1): μ₁ − μ₂ ≠ 0: States a real mean difference exists between the two populations.
One-Tailed (Directional) Test: Used when prior theory justifies a specific predicted direction of the difference.
Independent-Measures t Formula: t= (M1-M2)/s(M1-M2). Compares actual mean difference to expected error.
Estimated Standard Error: s(m1-M2). Measures the expected amount of error between a sample mean difference and the corresponding population mean difference.
Degrees of Freedom (df=n1+n2-2). Total df is obtained by combining the df from each of the two separate samples.
Pooled Variance (s²p): A weighted average of the two sample variances. (SS1 + SS2)/ (df1 + df2). Gives more weight to the larger sample.
Why pooling: When sample sizes differ, treating the two variances equally is biased. Pooling corrects this by letting the larger sample carry more weight.
Used in standard error: The pooled variance replaces individual sample variances in the standard error formula to produce an unbiased estimate
Three Assumptions: Conditions that must be satisfied for the independent-measures t test to be valid
Independence: Scores within each sample must be independent. No score influences another.
Normality: The populations from which samples are selected must be approximately normal (less critical with a large n)
Homogeneity of Variance: Both populations must have equal variances- this is required for pooling to be meaninful.
Hartley's F-Max Test: A simple test used to check whether two sample variances are significantly different from each other. This verifies the homogeneity assumption.
Effect Size Measures: Quantify the magnitude of the treatment effect independent of sample size.
Cohen's d: Standardized measure of the mean difference: d= (M1-M2)/ √s²p. This uses the pooled standard deviation as the denominator.
r² (Proportion of Variance): Measures how much of the total score variability is accounted for by the treatment effect. This is calculated with: t² / (t² + df)
Confidence Interval for μ₁ − μ₂: An interval estimate of the true population mean difference. This is constructed using the t statistic and standard error.
Link to Hypothesis Testing: If zero falls outside the confidence interval, H0 is rejected at the corresponding alpha level.
Interval width factors: Width depends on the confidence level chosen and the sample size. Larger samples produce narrower, more precise intervals.
Factors affecting the t statistic: Elements of the study design that influence how large or small t will be.
Sample Variance: Larger variance inflates standard error, produces a smaller t, and reduces the likelihood of rejecting H0.
Sample Size (n): Larger n reduces standard error, increases t, and increases power, but does not substantially affect Cohen's d.
Individual Differences: Person-to-person variability, such as differences in motivation/ability/experience, is a primary source of within-group variance and standard error.
Analysis of Variance (ANOVA): A hypothesis-testing procedure that evaluates mean differences among two or more treatment conditions using the F-ratio.
Why ANOVA instead of Multiple t Tests: ANOVA compares all group means simultaneously in a single test, avoiding the inflation of Type I error risk.
Experimentwise Alpha Level: The total accumulated probability of making at least one Type I error across all comparisons in an experiment. This grows with each additional test.
Testwise Alpha Level: The alpha level set for a single individual hypothesis test. This is kept at .05 or .01.
Factor: The independent (or quasi-independent) variable whose levels define the treatment groups being compared in ANOVA.
Levels of the Factor: The individual conditions or values that make up the factor.
Single-Factor Independent-Measures Design: ANOVA applied to one independent variable with separate groups of participants for each treatment level.
Null Hypothesis: H₀: μ₁ = μ₂ = μ₃ (all means equal). The alternative states that at least one mean differs. This does not specify which means differ.
Alternative Hypothesis: H1: Not all population means are equal. The alternative states that at least one mean differs. It does not specify which means differ.
Total Variability (SStotal): The overall spread of all scores in the study, partitioned into two components: between-treatments and within-treatments.
Between- Treatments Variance (SSbetween): Measures differences among the treatment group means. This reflects both treatment effects and sampling error.
Within-Treatments Variance (SSwithin): Measures variability of scores inside each treatment group where all participants received the same treatment. This reflects only random, unsystematic error.
F-Ratio= MSbetween/ MSwithin: The test statistic for ANOVA. This compares treatment-related variance to error variance.
Mean Square (MS): Variance estimate calculated as SS/df. This is used in place of the word "variance" in ANOVA.
F=1.00: Fail to reject H0. When H0 is true, numerator and denominator estimate the same error, so F should be near 1.
F>1.00: Reject H0. A large F indicates between-treatment differences exceed what random error alone would produce.
Error Term: MSwithin: The denominator of the F-Ratio. This measures only random, unsystematic variability within groups.
Key Notation: Symbols used throughout ANOVA calculations
T (Treatment Total): Sum of all scores within a single treatment condition
G (Grand Total): Sum of all scores across all treatment conditions in the entire study
k: Number of treatment conditions (levels of the factor)
N: Total number of scores in the entire study. N= k x n when the group sizes are equal.
Degrees of Freedom in ANOVA: df values partition just like SS values and are used to compute MS.
df total= N-1. Based on all scores in the study.
df between= k-1. Based on the number of treatment conditions.
df within= N-k. Sum of df from each individual treatment group.
η²=Eta Squared: Effect size measure for ANOVA: η²=SSbetween/ SStotal. This is the proportion of total variability explained by the treatment.
Interpretation: Like r², it indicates how much of the score variance is accounted for by the independent variable, independent of sample size
Distribution of F-Ratios: The sampling distribution of F when H0 is true. This is always positive and peaks near 1.00. It is right skewed.
Critical Value from F-Table: Determined by df between (numerator) and df within (denominator) and the chosen alpha level.
ANOVA Summary Table: Organizes SS, df, MS, and F values for all sources of variability in a single table.
Post Hoc Test: Additional hypothesis tests conducted after a significant ANOVA (with k greater than or equal to 3) to determine exactly which pairs of means differ significantly.
Pairwise Comparisons: Comparing group means two at a time.
Tukey's HSD Test: Computes a single minimum difference value (HSD= q√(MS_within/n)). Any mean difference exceeding HSD is significant. This requires equal n.
Scheffe Test: Most conservative post hoc test. This uses the full k from the original ANOVA in its df calculation, providing the greatest protection against Type I error.
ANOVA vs. Independent-Measures t: When comparing only two groups, both tests always yield the same statistical decision.
Mathematical Relationship: F= t². The F-ratio equals the square of the t statistic when k=2. df within in ANOVA equals df in the t test.
Correlation: A statistical technique that measures and describes the direction, form, and strength of the relationship between two variables.
Three Characteristics of a Correlation: Every correlation describes a relationship among three dimensions.
Direction (+ or -): Positive correlation= X and Y move in the same direction. Negative or inverse correlation= X increases as Y decreases.
Form: The pattern of the relationship. This is most commonly linear, where data cluster around a straight line.
Strength/Consistency (0 to + or - 1.00): Perfect correlation is close to + or - 1.00 (the data is not scattered). Zero correlation means there is no consistent trend. Values between represent the degree of scatter.
Scatter Plot: A graph with X on the horizontal axis and Y on the vertical axis. Each individual data point is represented by a single point, allowing the pattern of the relationship to be visualized.
Envelope: An oval drawn around all data points. A football-shaped envelope suggests that r is around .70. Fatter envelopes show weaker correlation and narrower show stronger.
Outliers: Individual data points far from the others that can dramatically distort the correlation value.
Pearson Correlation (r): The most common correlation. This measures the degree and direction of the linear relationship between two variables. The formula: r = SP / √(SS_x × SS_y)
Sum of Products (SP): Measures the co-variability of X and Y together. SP= Σ(X − Mx)(Y − My). This is analogous to SS but uses the product of two deviations instead of squared deviations.
Range: -1.00 to +1.00. It cannot exceed these bounds. Values outside indicate a calculation error.
Population Correlation (ρ): The Greek letter rho represents the true correlation in the population. r is the sample estimate.
Cautions: Several factors can distort or mislead interpretation of a correlation value.
Correlation does not equal Causation: A correlation shows that two variables are related but does not prove that one causes the other. A true experiment is needed to establish causation.
Restricted Range: If the data cover only a limited range of X or Y values, the correlation within that range may not reflect the true correlation for the full population.
Effect of Outliers: A single extreme data point can drastically inflate or deflate the correlation. The scatter plot should always be examined first.
Coefficient of Determination (r²): The squared correlation. This measures the proportion of variability in Y that is predicted or determined by its relationship with X.
Interpretation Standards: r²=.01=small effect. r²=.09=medium effect. r²=.25 or larger=large effect.
Link to Effect Size: r² functions as an effect size measure for correlation just as it does for t tests and ANOVA.
Testing H0: ρ = 0: Tests to determine whether a sample correlation represents a real (non-zero) relationship in the population or is simply due to sampling error.
t Statistic for r: t = r√(n−2) / √(1−r²). This is compared against critical t with df= n - 2.
df= n - 2: Two degrees of freedom are lost because both Mx and My must be known to compute the regression line.
Spearman Correlation (rs): Pearson formula applied to ranked data. This is used when variables are ordinal or when the researcher wants to measure the degree of monotonic relationships.
Monotonic Relationship: A relationship in which Y consistently increases or decreases as X increases, regardless of whether the pattern is linear.
Tied Scores Procedure: When scores are tied, average their rank positions and assign that mean rank to each tied score.
Special D Formula: rs=1- (6ΣD²)/(n(n²−1)), where D=difference between X rank and Y rank for each individual. This is used when there are no ties.
Point-Biserial Correlation: The Pearson Correlation is used when one variable is continuous (numerical) and the other is dichotomous.
Dichotomous Variable: A variable with only two possible values such as pass/fail, treatment/control, have/not have. Often coded as 0 and 1.
Link to Independent-Measures t: The point-biserial r and the independent-measures t are mathematically related: r² = t² / (t² + df). t measures the significance. r² measures the effect size.
Phi Coefficient (ϕ): This Pearson formula is applied when both variables are dichotomous. The sign is arbitrary. You interpret by using the coefficient of determination (r²)
Regression: The statistical procedure for finding the best-fitting straight line (regression line) to describe and predict the relationship between X and Y.
Regression Line Equation: Ŷ = bX + a
Slope (b): How much Y changes for every one-unit increase in X. b=SP/SSx. The sign matches the sign of the correlation.
Y-Intercept (a): The predicted value of Y when X=0. a=My-b(Mx).
Least-Squares Solution: The regression line minimizes the total squared error (Σ(Y − Ŷ)²) between actual Y values and predicted Ŷ values on the line.
Using the Regression Equation for Prediction: For any given X value, substitute Ŷ = bX + a to predict Y. The accuracy depends on the strength of the correlation.
Standardized Regression Equation: This is when X and Y are converted to z-scores first. zŷ = βzx. β (beta) = the Pearson r for the two variables
Caution: The regression equation should only be used for X values within the range of the original data. Predictions beyond that range are unjustified.
Standard Error of Estimate: This measures the standard or average distance between the actual Y values in the data and the predicted Ŷ values on the regression line. This is analogous to a standard deviation for prediction error.
SS residual: The sum of squared distances between the actual Y scores and predicted Ŷ values. SS residual=SSy(1 − r²)
df= n - 2: Two degrees of freedom are lost (one for Mx and one for My) when estimating the regression line.
Relationship to r: The stronger the correlation, the smaller the standard error of estimate. A near perfect r means data points cluster tightly around the regression line.
Analysis of Regression: Significance of the Equation: An F-ratio test (parallel to ANOVA) used to determine whether the regression equation predicts a significantly greater proportion of Y variance than expected by chance alone.
MS regression / MS residual: F=variance predicted by the regression equation divided by unpredicted (residual) variance.
H0 for Regression: β = 0: The null hypothesis states that the slope is zero, meaning that X does not predict Y in the population.
Equivalent to testing ρ = 0: When there is one X predicting one Y, the F-ratio for regression and the t test for correlation always yield the same statistical conclusion: F= t²