Psych stats test 4

test statistic

dividing a numerator that represents the difference between groups by a denominator that represents the variability within the groups—between-groups variability divided by within-groups variability

ANOVA

analysis of variability

-a hypothesis test typically used with one or more nominal independent variables (with at least three groups overall) and a scale dependent variable

F Statistic

-a ratio of two measures of variance: (1) between-groups variance

F\=between-groups variance/within-groups variance

A higher F statistic indicates less overlap among the sample distributions

evidence that the samples come from different populations

between-groups variance

-an estimate of the population variance based on the differences among the three (or more) means

-if there is a great deal of spread among several means

this suggests a difference exists among them

-reflects the difference between means that we found in our data

determine the variance among the three sample means

within-groups variance

-a weighted average of the variances within each sample

-an estimate of the population variance based on the differences within each of the three (or more) sample distributions

-reflects the difference between means that we'd expect just by chance

determine the variance within each of the three samples

and then take a weighted average of the three variances

between-groups vs. within-groups variance

If the between-groups variance (the numerator) is much larger than the within-groups variance (the denominator)

-use the F table to determine whether the ratio of the differences among our groups to the differences within each of our groups is extreme enough to reject the null hypothesis and conclude that a difference exists

F Table

-includes several extreme probabilities and the range of sample sizes

-includes a third factor

the number of samples

When to use z

t or F

t: 1) one sample

only mu is known

2) two samples

F: three or more samples (can be used with two samples)

descriptions of ANOVA

example: For a comparison of CFC scores across years in school

participants can be in only one level of the independent variable

1) indicates the number of independent variables

2) indicates whether the participants are in one condition (between-groups) or all conditions (within-groups)

example answer: one-way between-groups ANOVA

one-way ANOVA

a hypothesis test that includes one nominal independent variable with more than two levels and a scale dependent variable

-2 research designs: within-groups ANOVA or between-groups ANOVA

within-groups ANOVA

a hypothesis test in which there are more than two samples

-also called a repeated-measures ANOVA

between-groups ANOVA

-a hypothesis test in which there are more than two samples

3 assumptions for ANOVA

1) our samples are selected randomly

--necessary if we want to generalize beyond our sample (external validity)

2) the population distribution is normal

--examine the distributions of our samples to get a sense of what the underlying population distribution might look like

3) the samples all come from populations with the same variances

an assumption called homoscedasticity

homoscedastic

populations that have the same variance

-homogeneity of variance

check to see if the variances are similar (typically

when the largest variance is not more than twice the smallest) when we calculate the test statistic

*Note: When calculating an ANOVA

be sure to return to this step to indicate whether we meet the assumption of equal variances (assumption is met when the largest sample variance is not more than twice the amount of the smallest variance)

heteroscedastic

populations that have different variances

null vs. research hypothesis for ANOVA

Any combination of differences between means is possible when we reject the null hypothesis

H0:μ1 \= μ2 \= μ3 \= μ4.

No symbols used for H1 because only one has to be different from the rest

degrees of freedom for between-group variance

number of samples (or groups) minus 1

degrees of freedom for within-groups variance

find df for each sample (number of participants in particular sample minus one)

sum of each individual df

no negative F cutoff

F is based on estimates of variance instead of standard deviation or standard error in both the numerator and denominator

F cutoff

only one for a two-tailed test because F is a squared version of z or t

source table

presents the important calculations and final results of an ANOVA in a consistent and easy-to-read format

MS

mean square\=variance

df total

total number of people in the entire study minus one

or df between + df within

SS

sums of squares

SS between/df between\=MS between

SS within/df within\=MS within

SS total

total sum of squares

Sum(X-GM)^2

GM (grand mean)

the mean of every score in a study

sum(X)/Ntotal

SS within

the deviations are around the mean of each particular group

sum(X-M)^2

SS between

-estimate how much each group deviates from the overall grand mean

subtract grand mean from the mean of each group

this subtraction can be performed just once for each group

and the squared deviation score can be multiplied by the number of participants in the group

sum(X-GM)^2

SS recap

-for the total sum of squares

-for the within-groups sum of squares

we subtract the appropriate sample mean from every score

-for the between-groups sum of squares

we subtract the grand mean from the appropriate sample mean

--for the between-groups sum of squares

the actual scores are never involved in any calculations.

R^2

the proportion of variance in the dependent variable that is accounted for by the independent variable

R^2\=SSbetween/SStotal

effect size (R^2)

small\=0.01

medium\=0.06

large\=0.14

post-hoc test

a statistical procedure frequently carried out after we reject the null hypothesis in an analysis of variance; it allows us to make multiple comparisons among several means

allow us to determine which means are statistically significantly different from one another once we determine that there is a difference somewhere

Tukey HSD test

a widely used post-hoc test that determines the differences between means in terms of standard error; the HSD is compared to a critical value

involves (1) the calculation of differences between each pair of means and (2) the division of each difference by the standard error

HSD

-stands for "honestly significant difference" and indicates that we adjusted for the fact that we are making multiple comparisons

(M1-M2)/sM

sM (std error)\=sqrt(MSwithin/N)

--N in this case is the sample size within each group

with the assumption that all samples have the same number of participants

harmonic mean

weighted sample size (N')

N'\=N[groups]/(sum(1/N))

standard error with unequal sample sizes

When sample sizes are not equal

sM\=sqrt(MSwithin/N')

Independent-samples t tests

used when 2 groups have a between-groups design

-participants are in only one group or condition

paired-samples t tests

used when 2 groups have a within-groups design

-participants are in both groups or conditions

one-way within-groups ANOVA

repeated-measures ANOVA

used when there's just one nominal independent variable (type of beer)

the independent variable has more than two levels (cheap

reduce error due to differences between our groups (groups are identical on all of the relevant variables)

fourth assumption for one-way within-groups ANOVA

avoid order effects

ex: avoid tasting beers in same order across participants

4 degrees of freedom for one-way within-groups ANOVA

between-groups

fourth sum of squares for one-way within-groups ANOVA

for differences across participants

-called the subjects sum of squares or SSsubjects