stat methods - ch. 12,13,14,15

Analysis of Variance (ANOVA)

hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations)

Factor

In ANOVA, the variable (independent or quasi-independent) that designates the groups being compared.

Levels (levels of the factor)

The individual conditions or values that make up a factor

Two-factor design / factorial design

a study that combines two factors

Single-factor designs

studies that have only one independent variable (or only one quasi-independent variable)

single-factor, independent-measure design

Testwise alpha level

the risk of a Type I error, or alpha level, for an individual hypothesis test.

Experimentwise alpha level

when an experiment involves several different hypothesis tests, total probability of a Type I error that is accumulated from all of the individual tests in the experiment. Typically, the experimentwise alpha level is substantially greater than the value of the alpha used for any one of the individual tests.

Between-treatments variance

measures how much difference exists between the treatment conditions

Treatment effect

differences between treatments have been caused by this.

Ex. if treatments really do affect performance, then scores in one treatment should be systematically different from scores in another condition.

Within-treatments variance

provides a measure of how big the differences are when H₀ is true

F-ratio

• ratio of variance between treatments and variance within treatments

• helps determine whether any treatment effects exist

• F = variance btwn treatments ÷ variance within treatments. = differences including any treatment effects ÷ differences with no treatment effects

Error term

• its the denominator of the F-ratio for ANOVA

• provides a measure of the variance caused by random and unsystematic differences.

• when treatment effect is zero (H₀ is true), it measures the same sources of variance as the numerator of F-ratio, so value of F-ratio expected to be nearly equal to 1.00.

Mean square (MS)

• In ANOVA, customary to use this term in place of the term variance.

• mean of squared deviations

Distribution of F-ratios

all possible F values that can be obtained when the null hypothesis is true.

ANOVA summary table

• summary of all ANOVA calculations

  • SS, df, MS, F-value, and p-value

Eta sqaured

• η²

• percentage of variance accounted for by the treatment effect

• = SS_{between treatments} ÷ SS_total

Post hoc tests / posttests

additional hypothesis tests done after an ANOVA to determine exactly which mean differences are significant and which aren’t

Pairwise comparisons

comparing individual treatments two at a time

Tukey’s HSD test

• allows you to compute a single value that determines the minimum difference between treatment means that is necessary for significance

Name of value produced by Tukey’s HSD test:

Honestly significant difference or HSD

What can you conclude if the mean difference exceeds Tukey’s HSD?

that there is a significant difference between treatments

Scheffe’ test

uses an F-ratio to evaluate significance of difference between any two treatment conditions.

Numerator of the F-ratio:

an MS between between treatments

How is an MS between treatments calculated?

using only the two treatments you want to compare

What is the denominator of the F-ratio?

same MS_within that was used for the overall ANOVA

What are similarities between ANOVA and t tests?

both use sample data to test hypotheses about population means

Differences between ANOVA and t tests:

• t-tests are limited to situations in which there are only two treatments to compare

• ANOVA can be used to compare two or more treatments

• ANOVA provides researchers with much greater flexibility in designing experiments and interpreting results

What is the goal of the analysis done by ANOVA

to determine whether the mean differences observed among the samples provide enough evidence to conclude that there are mean differences among the three populations

What happens to the experimentwise alpha level as the number of separate tests increases?

it increases

A large value for the test statistic provides evidence that:

the sample mean differences (numerator) are larger than would be expected if there were no treatment effects (denominator)

Matrix

a set of numbers arranged in rows and columns so as to form a rectangular array

Cell

• an individual element or value located at the intersection of a row and a column in a matrix.

• each one represents a specific value identified by its position in the matrix

Main effect

mean differences among the levels of one factor

The mean differences between columns or rows describe:

the main effect for a two-factor study

Interaction

between two factors, it occurs whenever the mean differences between individual treatment conditions, or cells, are different from what would be predicted from the overall main effects of the factors

Simple main effects

the effect of one variable on one level of the other variable

Correlation

statistical technique that is used to measure and describe the relationship between two variables

Positive correlation

• two variables tend to change in the same direction

  • as value of X variable increases/decreases from one individual to another, Y variable also tends to increase/decrease

Negative correlation

• two variables tend to go in opposite directions

  • as X variable increases, Y variable decreases = inverse relationship

Direction of the relationship

sign of the correlation, positive or negative, describes the direction of the relationship

Envelope

line that encloses the data, and often helps you see the overall trend in the data.

When an envelope is shaped roughly like a football:

the correlation is around 0.7

Envelopes fatter than a football indicate what?

that correlations closer to 0

Narrow shaped envelopes indicate what?

correlations closer to 1.00

Pearson correlation

measures the degree and the direction of the linear relationship between two variables

Linear relationship

how well the data points fit a straight line

Sum of products (SP)

measures the amount of covariability between two variables

The value for SP can be calculated with either a:

definitional formula or a computational formula

Definitional formula

Computational formula

Outliers

extreme data points

Restricted range

observed data for a variable or variables is limited to a smaller portion of its potential range

coefficient of determination

• r² measures the proportion of variability in one variable that can be determined from the relationship with the other variable

Correlation matrix

results from multiple correlations are most easily reported in this table

Spearman correlation

result when Pearson correlation formula is used with data from an ordinal scale (ranks)

Monotonic

relationship when there’s a consistently one-directional relationship

Point-biserial correlation

used to measure relationship between two variables in situations in which one variable consists of regular, numerical scores, but the second variable has only two values.

Dichotomous variable

variable with only two values

Phi-coefficient

when both variables (X and Y) measured for each individual are dichotomous