Chapter 15, 16, 17 and Final Material

df for the numerator and denominator

n1-1 n2-1

df between

number of groups - 1 (k-1)

factorial experiment

one in which the effects of two or more factors or independent variables are assessed in one experiment In a factorial experiment, the conditions or treatments used are combinations of the levels of the factors. For example, in an experiment investigating the effects on sleep on two levels of exercise (light and heavy), carried out at two times of the day (morning and evening), there would be four treatments or conditions.

A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in how long they remain in the body before they are excreted or metabolized. The pharmacologist randomly assigned 18 subjects to three different groups. Each group is given a different type of psychoactive drug and the number of days the drugs remained at measurable levels in each subject is recorded. What statistical test is appropriate to analyze the results from this experiment?

one way ANOVA

A therapist measures the difference between two clients. If the therapist can say that Rebecca's score is higher than Sarah's, but can't specify how much higher, the measuring scale used must have been a(an)_______ scale.

ordinal

A personal trainer takes the weight of a group of dieters. The variable "weight" is measured on what type of scale?

ratio

MS within

since an assumption of the analysis of variance is that the independent variable affects only the mean and not the variance of each group, the within-groups variance estimate does not change with the effect of the independent variable

You conduct an experiment to determine if the cingulate cortex is involved in learning tasks involving choice behavior. Twenty rats with lesions of the cingulate cortex are tested in a two choice Y-maze with the correct arm of the maze being randomly determined from trial-to-trial. Previous research with a large number of rats on this task has shown that the mean number of trials learn the task is 20 trials. The results of the experiment show a mean of 22 trials with a standard deviation of 5.4 for the lesioned rats to learn the task. What statistical test is appropriate?

single sample t test population parameters are unknown

df within

N-k (total # of observations - total # of groups)

Size of effect using ωˆ 2

0.01-0.05 = Small 0.06-0.13 = Medium greater than 0.14 = large

Assumptions Underlying χ2

1. Independence exists between each observation in the contingency table. 2. Sample size is large enough so that the expected frequency in each cell is at least 5 for tables where r or c is greater than 2. 3. If table is 1 x 2 or 2 x 2 than each expected frequency should be at least 10. 4. χ2 can be used with nominal, ordinal, interval or ratio data.

Assumptions underlying use of ANOVA

1. Populations from which the samples were taken are normally distributed. 2. Samples are drawn from populations of equal variances. • Homogeneity of variance assumption • ANOVA is a robust test. • F is robust if sample sizes are equal. • F is minimally affected by violations of population normality

Power of ANOVA

1. Sample size. Increasing sample size increases power. 2. Real effect of the independent variable. The larger the real effect of IV, the higher the power. 3. Sample variability. The lower the sample variability, the higher the power to detect a real effect. power varies directly with N and real effect of independent variable, and inversely with within-group variability.

Assumptions underlying two- way ANOVA

1. The populations from which the samples were taken are normally distributed. 2. Homogeneity of variance Violations of Assumptions. • Two-way ANOVA is robust with regard to violations of these assumptions, provided the samples are of equal size.

A correlated groups t test is conducted in which the same 9 subjects are tested during a baseline period and after a specific training. Using ɑ = 0.052 tail, what is the value of tcrit?

2.306

A one-way ANOVA is conducted with a total of 18 subjects. Each subject is randomly assigned to one of three groups. Using ɑ = 0.05, what is the value of Fcrit?

3.68

interaction effect

An interaction effect occurs when the effect of one factor is not the same at all levels of the other factor.

Fobtained vs Fcrit

If Fobt ≥ Fcrit, reject H0 If Fobt < Fcrit, retain H0

significance

If the obtained probability ≥ α, then we reject H0; if not, we retain H0

df total

N-1 (N is total # of observations)

• Interaction effect

Occurs when the effect of one factor is not the same at all levels of the other factor • Each Fobt value is evaluated against Fcrit (as in one-way analysis) If Fobt ≥ Fcrit, there is sig. interaction effect

Pearson vs. Spearman correlation

Pearson correlation tests for the strength of the association between two continuous variables Spearman correlation tests for the strength of the association between two ordinal variables (does not rely on the assumption of normally distributed data)

Relationship between Fobt and the independent variable:

Since Fobt = MSbetween/Mswithin - as the IV effect increases, Fobt increases.

chi square

Tests for the strength of the association between two categorical variables

When to use sign test

Tests if two related variables are different—ignores the magnitude of change, only takes into account direction

hypotheses of ANOVA

The alternative hypothesis used in the analysis of variance is nondirectional. It states that one or more of the conditions have different effects from at least one of the others on the dependent variable. The null hypothesis states that the different conditions are all equally effective, in which case the scores in each group are random samples from populations having the same mean value. the analysis of variance assumes that only the mean of the scores is affected by the independent variable, not the variance

main effect

The effect of factor A (averaged over the levels of factor B) and the effect of factor B (averaged over the levels of factor A).

Relationship Between ANOVA and the t Test

When a study involves just two independent groups and we are testing the null hypothesis that μ1 = μ2, we can use either the t test for independent groups or the analysis of variance. In such situations, it can be shown algebraically that t2 = F.

planned comparisons vs post hoc

With planned comparisons, we do not correct for the higher probability of Type I error that arises due to multiple comparisons, as is done with the post hoc methods. This correction, which we shall cover in the next section, in effect makes it harder for the null hypothesis to be rejected. Because planned comparisons do not involve correcting for the higher probability of Type I error, planned comparisons have higher power than post hoc comparisons. In doing planned comparisons, the t test for independent groups is used. We could calculate tobt in the usual way. Planned comparisons are the most powerful of the multiple comparison tests.

how is the t test robust

a test is robust if violations in the assumptions do not greatly disturb the sampling distribution of its statistic. Thus, the t test is robust regarding the violation of normality in the population. Even though, theoretically, normality in the population is required with small samples, it turns out empirically that unless the departures from normality are substantial, the sampling distribution of t remains essentially the same. Thus, the t test can be used with data even though the data violate the assumptions of normality

Classifying subjects on the basis of sex is an example of using what kind of scale?

a. nominal

sampling distribution of F gives

all the possible F values along with the p(F) for each value, assuming sampling is random from the population First, since F is a ratio of variance estimates, it never has a negative value (s12 and s22 will always be positive). Second, the F distribution is positively skewed. Finally, the median F value is approximately equal to 1. there is a different curve for each combination of df and df2

F-test (ANOVA)

analysis of variance - analyzes differences between multiple conditions simultaneously - one test instead of multiple tests

F test is appropriate when

appropriate in any experiment in which the scores can be used to form two independent estimates of the population variance. One quite frequent situation in the behavioral sciences for which the F test is appropriate occurs when analyzing the data from experiments that use more than two groups or conditions Using the F test allows us to make one overall comparison that tells whether there is a significant difference between the means of the groups.

A researcher believes that if muscle tension is reduced, tension headaches would decrease or disappear. He designs an experiment using nine subjects that experience tension headaches. Subjects keep daily logs of the number of headaches during a two-week baseline period. Subjects are then trained to lower their muscle tension using a biofeedback device. Once subjects learn to lower their muscle tension, they keep a two-week log of the number of headaches experienced. What statistical test is appropriate to analyze the results from this experiment?

correlated groups t test

degrees of freedom for ANOVA

df for MSwithin = N - k df for MSbetween = k - 1 dftotal = N - 1

Chi-square should not be used if _________.

fe is below 5

when to use the t test for independent groups

if you have TWO groups of subjects AND they are different individuals that participate in each condition (aka independent groups) require that the population scores be normally distributed when the samples are small requires that the population variances be equal

when to use the single sample t test

if you have a single sample but do not know the population standard deviation requires that we specify the mean and standard deviation of the null-hypothesis population, as well as requiring that the population scores must be normally distributed for small Ns. The t test for single samples has the same requirements, except that we don't specify σ

when to use chi square

if you have nominal data (frequency of each occurrence) or if you have serious violations of an assumption and you cannot use parametric test

When to use one-way ANOVA

if you have three or more groups of subjects but only one independent variable

When to use a Two-Way ANOVA

if you have two independent variables

In a two-way ANOVA, if there is a significant interaction between Factor A and Factor B, which of the following may be true?

the effect of Factor A is not the same at all levels of Factor B and/or The effect of Factor B is not the same at all levels of Factor A

A main effect for variable A means that _________.

the effect of variable A is significant when averaged over all levels of variable B

F obtained is

the ratio of two independent variance estimates of the same population variance σ2

An experiment was conducted to assess the effects of a minor tranquilizer on a performance task at different levels of stress. The levels of stress examined were moderate and high and the levels of tranquilizer were none and moderate. What statistical test would be appropriate to analyze this experiment?

two way ANOVA

when to use the t test for correlated groups

use if you have a repeated measures design; SAME individuals participate in both of your conditions or the participants are specifically matched on a certain characteristic/(s) require that the population scores be normally distributed when the samples are small

when to use z test

use if you have a single sample and know the population parameters both population men and standard deviation requires that we specify the mean and standard deviation of the null-hypothesis population, as well as requiring that the population scores must be normally distributed for small Ns

An automotive engineer believes that the engine she designed will be a gas saver. A large number of tests on engines of the old design yielded a mean gasoline consumption of 27.5 miles per gallon, with a standard deviation of 5.2. Fifteen new engines are tested. The mean gasoline consumption is 29.6 miles per gallon. What statistical test is appropriate?

z test

A posteriori or post hoc comparisons

• Comparisons NOT planned in advance of the experiment • NOT based on specific predictions or prior research • Conducted after the researcher looks at the data and decides which groups to compare • OR researcher wants to do all possible comparisons • We correct for inflated probability values that occur when doing multiple comparisons • Maintain Type I error rate at alpha • Tukey HSD (Honestly Significant Difference) test: Maintains Type I error rate at alpha when controlling for all possible comparisons between pairs of means • Scheffé test: Maintains Type I error rate at alpha when controlling all possible comparisons (not just pairwise mean comparisons) • Scheffé test is less powerful than Tukey HSD

size of effect using η2

• Eta squared is another method for estimating size of effect in one-way independent groups ANOVA • Similar to omega squared • Estimate is usually larger than true size of effect

• Two-way ANOVA:

• Evaluate the effect of two IV and the interaction between them Experiment investigating the effects on sleep of two levels of exercise (light & heavy) at two times of day (morning & evening) • Calculate Fobt value for: • Variable A:If Fobt ≥ Fcrit, there is sig. main effect • Variable B: If Fobt ≥ Fcrit, there is sig. main effect • Interaction between A and B If Fobt ≥ Fcrit, there is sig. interaction effect One main difference is that in the two-way ANOVA we are often evaluating pairs of row means or column means rather than pairs of group means.

Sampling distribution of F

• Gives all the possible F values along with p(F) for each value, assuming sampling is random from the population. • F distribution varies with degrees of freedom • F distribution has two values for df • df for numerator = dfbetween • df for denominator = dfwithin F is always positive • Because it is a ratio of variance estimates (s12 and s22 will always be positive) • F distribution is positively skewed (see p. 403) • When n's are equal, median value of F = 1 • F distribution is a family of curves for each combination of df

• F values and real effect of the independent variable:

• If H0 is true, Fobt is expected to equal 1. • If Fobt < 1, you will retain H0

Within-groups variance estimate, MSwithin

• It tells use how large the differences are between the scores within each group and the group mean • Same as sw2 used for t test. • estimate of variance squared

Relationship between MSbetween and the independent variable

• MSbetween increases with size of effect of independent variable.

Chi-square (χ2)

• Nominal data • Discrete, mutually exclusive categories • Count frequencies in each category • Single-variable experiments Test of Independence between two variables • Experiments with two categorical variables Preference for Different Brands of Beer Grading system preference and major Political Affiliation and Attitude • Null hypothesis: attitude toward the bill and political affiliation are independent • Alternative hypothesis: Republicans and Democrats differ in their attitudes toward the bill

One-way ANOVA

• Only one IV • Two or more levels (conditions) of IV • Subjects are randomly assigned to a condition • Factorial experiment: • Effects of two or more IV (or factors) are assessed simple randomized-group design.

Chi square characteristics

• Sampling distribution is family of curves • Varies with df (similar to t distribution) • At lower df, curves are positively skewed (see page 486) • For experiments with one variable: • df=k-1 • k = number of groups or categories • The larger the discrepancy between the observed and expected results the larger the value of χ2obt • More unreasonable H0 is. • If χ2obt > χ2crit, reject H0 • Χ2 is a nondirectional test

• Nonparametric tests

• Sign test • Minimal requirements • "distribution-free tests"

• Relationship between MSwithin and the independent variable:

• Since IV affects only the mean and not the variance; MSwithin does not change.

A priori or planned comparisons

• Specific comparisons planned in advance of the experiment • Based on specific predictions or prior research • We do not correct for higher probability of Type I error • Higher power than post hoc comparisons

F test and Analysis of Variance (ANOVA)

• Two-group study (control & experimental group) • Two groups may not be sufficient for clear interpretation of the findings • Want to test several values of independent variable • Use experimental design with more than two groups • Example: "Hormone X and sexual behavior" study • Administer various dose levels (or concentrations) to different groups of subjects • If you do not know what level will be effective, this increases the possibility of a positive result If you do multiple t tests, the probability of making Type I error will increase for each comparison!

F test and Analysis of Variance (ANOVA) Summary points:

• Use: 1. To analyze data from experiments that employ more than two groups (or conditions). 2. Instead of many pairwise t tests in order to hold the probability of making a Type I error at alpha. F test allows us to make one overall comparison that tells us if there is a significant difference between means of the groups • Used: • Independent groups designs • Repeated measures designs • Independent groups design: • simple randomized group design • or the one-way analysis of variance • In the independent groups design, there are as many groups as there are levels of the independent variable. • Example: if a study is examining the effect of IV, then conditions would be different levels of the IV used. • Each group would receive a different level of IV • Hypothesis testing: • H1 is nondirectional • H0 states that different conditions are all equally effective • ANOVA assumes that IV affects mean of scores, but not variance

Variables to evaluate F obtained

• k = number of groups in the experiment N = total sum of groups dfbetween = k - 1 dfwithin = N - k dftotal = N - 1 Degrees of Freedom Numerator = dfbetween Degrees of Freedom Denominator = dfwithin

• Parametric tests

• z test • t test • F test (ANOVA) 1. Parametric inference tests are robust 2. Parametric inference tests are more powerful than nonparametric tests • Also, more versatile • e.g., with ANOVA you could test two, three, four, or more variables and their interactions! • General rule: use parametric tests when possible