Understanding ANOVA and Hypothesis Testing

ANOVA

Statistical method for comparing multiple means.

Hypothesis Testing

Procedure to determine if a hypothesis is true.

Null Hypothesis (H0)

Assumes no effect or difference exists.

Alternative Hypothesis (H1)

Suggests a significant effect or difference exists.

Type I Error

Rejecting H0 when it is actually true.

Type II Error

Failing to reject H0 when it is false.

Alpha (⍺)

Probability threshold for Type I error, usually 0.05.

t Test

Used to compare means of two groups.

Experimentwise Error Rate

Cumulative probability of Type I error across tests.

Testwise Error Rate

Probability of Type I error for a single test.

William Gossett

Statistician known for developing the t test.

Probability of Rolling a 1

Chance of rolling a '1' on a die.

Probability After Four Rolls

Chance of rolling a '1' at least once.

Number of Comparisons (C)

Calculated as C = k(k-1)/2.

k (means)

Number of groups being compared in a test.

Significant Result

Outcome indicating a likely true effect.

Replications Fail

Subsequent tests do not support initial significant findings.

k

Number of groups compared in ANOVA.

C

Total number of comparisons made.

⍺exp

Approximate experimentwise alpha level.

ANOVA

Statistical method for comparing group means.

Type I error

Incorrectly rejecting a true null hypothesis.

Type II error

Failing to reject a false null hypothesis.

Sir Ronald Fisher

Pioneer of ANOVA statistical methods.

Dependent variable

Outcome variable measured in an experiment.

Independent variable

Variable manipulated to observe effects.

Logistic regression

Used for binary categorical dependent variables.

Linear regression

Analyzes relationship between numerical variables.

Contingency table analysis

Examines relationships between categorical variables.

t-test

Compares means between two groups.

One-factor ANOVA

Compares means across multiple groups with one factor.

MSB

Mean squares between groups in ANOVA.

MSW

Mean squares within groups in ANOVA.

F statistic

Ratio used to determine group mean differences.

Homogeneity of variance

Assumption that group variances are similar.

F distribution

Distribution of F statistic under null hypothesis.

Null hypothesis (H0)

Assumes no difference between group means.

Alternate hypothesis (H1)

Assumes at least one group mean is different.

Driving ability assessment

Measured from 1 (worst) to 10 (best).

Rejection region

Area in F distribution for rejecting null hypothesis.

Mean

Average value of a data set.

Variance

Measure of data spread, squared standard deviation.

Standard Deviation

Square root of variance, indicates data dispersion.

Grand Mean

Overall average of all groups combined.

MSB

Mean squares between groups, variance among group means.

MSW

Mean squares within groups, error term for variance.

Degrees of Freedom

Number of independent values in a calculation.

F-Statistic

Ratio of variances, used in ANOVA tests.

ANOVA

Analysis of variance, compares means across groups.

Placebo Group

Control group receiving no treatment.

100 mg Group

Group receiving a 100 mg dosage.

250 mg Group

Group receiving a 250 mg dosage.

500 mg Group

Group receiving a 500 mg dosage.

dftotal

Total degrees of freedom in the study.

dfbetween

Degrees of freedom for group differences.

dfwithin

Degrees of freedom for individual differences.

Fcritical

Threshold value from F-distribution table.

Fcalc

Calculated F-value from the data.

p-value

Probability value indicating statistical significance.

H0

Null hypothesis, no effect or difference expected.

Effect Size

Magnitude of difference between groups.

Error Term

Variance within groups, used in F calculations.

ANOVA

Analysis of variance for comparing group means.

Sum of Squares

Total variance calculated from group data.

df

Degrees of freedom in statistical tests.

Mean Squares

Average variance per degree of freedom.

F-statistic

Ratio of variance between groups to within groups.

p-value

Probability of observing data under null hypothesis.

Between Groups

Variance attributed to group differences.

Within Groups

Variance attributed to individual differences.

Total Variance

Combined variance from all sources.

Eta Squared (η²)

Proportion of total variance explained by independent variable.

Omega Squared (ω²)

Adjusted measure of effect size for ANOVA.

Independent Samples t-test

Compares means of two independent groups.

Cohen's d

Effect size measure for differences between two means.

Residuals

Differences between observed and predicted values.

Treatment

Independent variable in ANOVA context.

Control Group

Group not exposed to treatment.

Experimental Group

Group exposed to treatment or intervention.

Grand Mean

Overall average of all group means.

t-statistic

Ratio of difference between group means to variability.

Critical t-value

Threshold for rejecting null hypothesis in t-test.

t-distribution

Probability distribution for t-statistics.

k (levels)

Number of groups or categories in ANOVA.

MSB

Mean squares between groups; variance among group means.

MSW

Mean squares within groups; error term variance.

Degrees of Freedom

Number of independent values in analysis.

F-statistic

Ratio of MSB to MSW; tests group differences.

Effect Size

Measure of the strength of a phenomenon.

Eta Squared (η²)

Proportion of variance attributed to group differences.

F-distribution Table

Table used to find critical F values.

Critical Value

Threshold for rejecting null hypothesis in ANOVA.

Null Hypothesis (H0)

Assumption that all group means are equal.

Alternative Hypothesis (H1)

Assumption that at least one group mean differs.

t-test

Statistical test comparing means of two groups.

ANOVA

Analysis of variance; compares means across multiple groups.

p-value

Probability of observing results under null hypothesis.

Variance

Measure of data spread around the mean.

Sum of Squares (SS)

Total variation in data; used in variance calculations.

Control Group

Group not exposed to experimental treatment.