PSYC 210 Quiz 5

Factorial ANOVA hypothesis

-Factor A: H0: alpha1 = alpha2 = 0 H1: Not H0 -Factor B: H0: beta1 = beta2 = 0 H1: Not H0 -Interaction H0: alphabeta11 = alphabeta12

A Priori Comparisons

-If we wanted to test whether workbook was different than the control, we would... carry out a planned contrast to get an F statistic with DF to get this: Thus, we fail to reject — we do not have enough evidence to argue that 5.2 is different than 4.0 because the critical F does not surpass the critical value with 1 and 16 degrees of freedom on the table -do not have enough evidence to suggest that students who just use textbook puts students in a different population than just simple lecture.

One Way ANOVA Hypothesis

-Null hypothesis: MS between and MS within are just error variance fail to reject — no group differences, any differences are due to chance and that all groups come from same underlying population -Null hypothesis is false: MS between = group differences + error MS within = error at least one of these means come from a different population — group matters (p was less than .05 — instructional type matters, scores on stats test differ as a function of the group you are in — SINGLE OMNIBUS EFFECT)

two methods for testing particular means

-Planned contrasts (a priori): planned ahead of time, based on a theory or question of interest (we think that the best way to teach is a combination — so we set up a test that tests just that — is the combination more effective than the 2 means) -Post hoc tests: dont have a specific hypothesis or want to keep our options open— after the fact to see if any of the means differ from any of the others

More than 2 factors in ANOVA?

-Works exactly the same way -SS between groups gets broken up into more parts - A, B, C, A x B, A x C, B x C, A x B x C -Partitioning the variance is key

Factorial ANOVA Example - sleep and testing

-does sleep impact test performance? What about test difficulty? -factorial ANOVA is important when you have a "when" question - when is something more likely to affect the other -create a factorial design with 2 factors: sleep (8 hrs vs 4 hrs) and test difficulty (easy vs hard) -4 conditions in our experiment (which is why we call this a 2x2 factorial bc there are two factors with two levels) -can see how two variables interact to see a particular pattern of results (what does the data look like) -marginal means: allow us to average across to get main effect of test difficulty and sleep to see how much the specific part matters, don't care about the other variable at that point

Main Effects

-first, look at the main effect of the variable independently - isolate the effect of a variable -the main effect is the effect of one factor, ignoring (or collapsing over) the other by calculating an average -have as many main effects as you have independent/dependent variables (2 sleep variables and two test difficulty variables)

All Possible Comparisons

-see if any means differ from any of the other -sometimes we find something that we didn't expect to find (like 2-tailed tests where the results can be completely flipped) -However, it can increase the chance of type 1 error -for example, if we did 6 comparisons... youre compounding your type 1 error rate of .05 (called family-wise type 1 error) to get a new alpha level of over .25 — use corrections to mitigate this threat or planned contrasts to look at the tests/comparisons you are interested in (stomach pain — small incision or take out all organs and check everything (planned > systematic — but sometimes your judgement call that is best might not be the best, but it might be better than overcorrection)

interaction effect

-the effect of one factor is different depending on the level of the other factor (or factors) -Size of the effect of one factor changes or varies as a function of the other variable or factor -ex: effect of amount of sleep differs depending on whether we look at an easy or difficult test.(when you get an easy test, theres not much difference, but when you get a hard test, there is a difference - there can be no main effects but an interaction)

Factorial design

-two or more independent variables (factors) -completely crossed: each level of one factor is combined with each level of every other factor (same logic as punnet squares)

Post Hoc Tests

-when want to leave no stone unturned -Fisher's LSD Test: least significant difference test — criticized for being too liberal, likely to find significant differences that might not actually be there -Tukey Test: honestly significant difference test, in many papers today (gold standard) -Scheffe Test: more conservative -Bonferroni Correction: MOST conservative, probably will never find anything — not really a post hoc test, but really just raises alpha value to protect against type 1 error -all tests differ in how conservative they are

Assumptions of factorial ANOVA

1) Interval or ratio (continuous) outcome (dependent variable) 2) Categorical independent variable 3) Homogeneity of variance 4) Independence of observations can use variables that's continuous and can handle categorical variables as well

If I have a factorial design examining the impact of sleep and test difficulty on performance, how many factors are present?

2

If I used a 2x3 design, how many cells or conditions would I have?

6

interactions and graphs

A. No interaction, no main effects B. No interaction, main effect for sex (males > females) C. No interaction, main effect for reinforcement (reward > punishment) D. No interaction (males > females for both reward and punishment), main effect for sex (males > females) and for reinforcement (reward > punishment) E. Ordinal/spreading Interaction (males > females when rewarded), main effect for sex (males > females) AND reinforcement (reward > punishment) F. Disordinal/crossover Interaction (females > males when rewarded, males > females when punished), no main effect

Broadly speaking, ANOVA allows us to compare

ANOVA uses the F test to test hypotheses — ANOVA tests variance (compares different sources of variances to form an F ratio that we can test against a critical value)

Factorial ANOVA

An Analysis of Variance used when there are two or more independent variables. When there are two, the ANOVA is called a Two-Way ANOVA, three independent variables would use a Three-Way ANOVA, etc. (Def. from allpsych.com)

What test semantic is used in ANOVA

F

Multiple Regression vs. ANOVA

In ANOVA, the independent variable(s) are always categorical. In this way, we end up comparing group means, where the groups are formed by the values of the categorical variables. We call categorical variables "factors". The focus of ANOVA is on comparing group means. We usually don't talk about prediction in ANOVA. With factorial ANOVA, we can examine not only the main effects of the variables but also the effects of interactions between the variables on the outcome. In regression, the independent variables can be continuous or categorical, but in PSYC 210, we've only talked about regression in the context of continuous independent variables. The focus of regression has been on the prediction of the dependent variable by the independent variables. The output of regression analysis is a regression line the relates the expected value of the dependent variable to the independent variables, and this equation can be used to make predictions of the dependent variable from given values of the independent variables.

simple effects tests

Tell us which means differ Uses F test logic (+ keep degrees of freedom) Using multiple t-tests (- lose degrees of freedom)

How would you interpret an interaction effect between sleep and test difficulty on test performance?

The effect of sleep on test performance depends on test difficulty

For the previous example: I have a significant main effect of sleep. What does this mean?

There is a difference in test performance as a function of sleep averaging across test difficulty

between-groups variance

an estimate of the population variance, based on the differences among the means

within-groups variance

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

Analysis of Variance (ANOVA)

hypothesis test typically used with one or more nominal (and sometimes ordinal) independent variables (with at least three groups overall) and a scale dependent variable = the larger the between groups variability and the smaller the within groups variability, the larger the F statistic value, and the more likely you are to reject the null hypothesis

significant omnibus difference problem

if there is or are significant differences between groups relative to what we'd expect to see by chance alone, a significant F value is great but it just tells us at the very least that one mean is not like the other (maybe 1, 2, or all means come from different populations) -simply know that null hypothesis is not viable

r squared

proportion of variance in the dependent variable that is accounted for by the independent variable = formula - r squared = SS(between)/SS(total) = conventions - small => 0.01 - medium => 0.06 - large => 0.14

F statistic

ratio of two measures of variance: between-groups variance, which indicates differences among sample means, and within-groups variance, which is essentially an average of the sample variances = significance is determined when the between group variance is much larger than the within groups variance, producing a large value = if the between groups variance and within groups variance are practically the same, with a value coming out to around one, then the means are not really different from one another

Factorial ANOVA Sum of Squares

sums are still additive, more precise tool in many ways.

grand mean

the mean of every score in a study, regardless of which sample the score came from

What does a Factorial ANOVA explore?

whether it is one of the IVs that explains the variance (or difference) on the DV OR if it's the interaction of the two (or more) IVs that explain the variance on the DV

What do post hoc tests tell us?

which means differ

Oneway ANOVA is useful when

you are examining for differences on a continuous outcome and you have more than two groups