Exam 5 - general information

SStotal =

SStrt + SSerr

The SStrt may not be called “trt” and can instead by any?

Independent variable

If you see or SStot, it means?

SStot

If you see, SSwithin or SSresidual, it means?

SSerr

SSIV name​,SSbetween​, SSmodel​ it means?

SStrt

What does N mean?

Sample size of one group

How to find N?

Count the people in that group.

What is Nt?

Sample size per treatment (group)

How to find Nt?

Same as N only if all groups are equal in size.

What is Ntot?

Total sample size

How to find Ntot?

Add everyone across all groups.

What is K?

Number of treatment groups (levels of the IV)

How to find K?

Count the groups/levels.

________: Used to find out if there any significant differences among three or more population means.  

One-Way ANOVA

Q: What question does a one-way ANOVA answer?

Do the populations represented by the samples have the same population mean (μ)? In other words, are all the population means equal, or is at least one different?

Instead of conducting a t-test and calculating a t-obtained value, ANOVA uses the f sampling distribution to compare ___-obtained value to __-critical value

f

Why use an ANOVA over multiple T-tests?

If we compute multiple t-tests:

• Use up degrees of freedom. 

• Inflated Type 1 Error Rate: Increase chances of finding significance when it isn’t actually there.

_____: Sum of squared deviations from the mean

Sum of Squares (SS)

_______: Variance; Sum of squares divided by its respective degrees of freedom 

Mean Square (MS)

• _______: Mean of all scores across all groups

• Grand Mean (X̅_G)

• _________: f-obtained value

• F-test:

____: Number of treatments in the experiment; Number of levels of the independent variable (IV)

K

The total variability is also called the?

SStot

Degrees of Freedom are always ?

Positive

______ ____ is simply a sum of squares divided by its degrees of freedom.

Mean Square

_________ = SS_treat/df_treat

MS_treat

_______ = SS_error/df_error

MS_error

Fc > Fo    _____ the null

Accept

Fc < Fo    _____ the null.

Reject

_______ Test: Pairs each sample mean with every other mean

Tukey HSD

in a Tukey HSD, _____ comparisons tells us if one sample mean is significantly different from the other.

pairwise

• HSD_c > HSD_o  ________ difference between means.

• No significant

• HSD_c < HSD_o  __________ difference between means.

• Significant

Assumptions of ANOVA

Normality, Homogeneity of Variance, Random Assignment

_______: Dependent variable is assumed to be normally distributed in the target populations.

Normality

• ______________: Variances of the dependent variable scores for each of the populations are assumed to be equal. 

• Homogeneity of Variance

SPSS can run a test of homogeneity. 

__________: Distributes the effect of extraneous variables equally across all groups. Allows us to conclude f-obtained is significant; that differences are too large to be due to chance.

Random Assignment

If all 3 assumptions of an ANOVA are not met, use a _____ test instead. 

nonparametric

____________ will also have just one independent variable (IV) with three or more levels. However, repeated measures means we will assess the same participants over time, use natural triplets, or matched groups.

Repeated Measures ANOVA

In repeat measures, Independent Variable (IV) = ______ 

• Factor

In repeat measures ANOVA, Source Table will now include _____________ and __________

between-treatment variance, error variance.

__________is a more sensitive measure of any differences that exist among the treatments because the variability produced by differences among the participants has been removed.

F statistic

• _____ = total number of scores 

• _____ = number of scores that receive one treatment

• _____ = number of treatments

• N_tot = total number of scores 

• N_t = number of scores that receive one treatment

• N_k = number of treatments

Cautions of an ANOVA?

Practice/Carryover Effect, Fixed Effects Model, ANOVA Assumptions

_____________ If one level of independent variable (IV) continues to affect the participant’s response in the next treatment condition. 

Practice/Carryover Effect:

__________: The level of IV must be chosen by the researcher instead of being selected randomly

Fixed Effects Model

_____________: The researcher needs to be mindful of meeting assumptions of parametric tests (normality, homogeneity of variance, and random assignment).

ANOVA Assumptions

Whenever you see F_{(a, b)} The first number (a) represents ________ (_____ groups) → ____ - ___

Treatment; between, K - 1

Whenever you see F_{(a, b)} The second number (b) represents ________ (_____ groups) → ____ - ___

Error; within, N - K

In a one-way ANOVA, the two degrees of freedom correspond to two different sources of variation. What are they?

Treatment (Between Groups); Error (Within Groups)

F (df_numerator, df_denominator) = F (_____, _____)

df_treat, df_error

If F < F.₀₅, then p ?

>.05

• If F_.05 ≤ F < F_.01, then p ?

< .05

• If F ≥ F_.01, then p?

< .01

If three sample means come from a population with μ = 100 and one sample mean comes from a population with μ = 50, how does the between-treatments variability compare to if all four samples came from populations with μ = 100?

A: The between-treatments variability is larger because one group mean is much farther from the others, increasing differences among the treatment means.

Q: What does the between-treatments estimate of variance (MSₜᵣₜ) measure?

A: Variability between the group means. It reflects treatment effects plus random error.

Q: What does the within-treatments estimate of variance (MSₑᵣᵣ) measure?

A: Variability within each group, caused by random error or individual differences—not the treatment.

Q: What is the main difference between the between-treatments and within-treatments estimates of variance?

A: Between-treatments compares differences among group means (treatment effect + error). Within-treatments measures differences within each group (error only).

Q: What are the two assumptions of a one-way ANOVA?

A:

1. The populations are normally distributed.

2. The populations have equal variances (homogeneity of variance).

Q: Why is the assumption of normality important in ANOVA?

A: It helps ensure the F-test is valid and the sampling distribution follows the expected pattern.

Q: What does homogeneity of variance mean?

A: All populations being compared have approximately the same variance.

Q: What is the advantage of random assignment of participants to treatments?

A: It helps make the treatment groups equivalent by evenly distributing participant differences across groups, reducing bias and confounding variables.

Q: Which estimate of variance should contain only random error?

A: The within-treatments estimate of variance (MSₑᵣᵣ).

Q: If the treatment has a real effect, which estimate of variance is expected to increase?

A: The between-treatments estimate of variance (MSₜᵣₜ) because the treatment causes the group means to differ.

Q: An ANOVA produced F = 3.12 with df = 3, 36.

• How many treatment groups are there?

• What are the critical values at α = .05 and α = .01?

• What conclusion should you reach?

• K = 4 (because df_treatment=K−1

• F₀.₀₅ ≈ 2.87

• F₀.₀₁ ≈ 4.38

• Since 3.12 > 2.87 but 3.12 < 4.38, p < .05 but p > .01.

• Reject H₀ at .05, but not at .01.

Q: An ANOVA produced F = 1.94 with df = 2, 27.

• How many treatment groups are there?

• What conclusion should you reach?

• K = 3

• F₀.₀₅ ≈ 3.35

• F₀.₀₁ ≈ 5.49

• Since 1.94 < 3.35, p > .05.

• Retain H₀.

Q: An ANOVA produced F = 6.81 with df = 4, 20.

• How many treatment groups are there?

• What conclusion should you reach?

A:

• K = 5

• F₀.₀₅ ≈ 2.87

• F₀.₀₁ ≈ 4.43

• Since 6.81 > 4.43, p < .01.

• Reject H₀ at both .05 and .01.

Q: An ANOVA produced F = 2.30 with df = 5, 30.

• How many treatment groups are there?

• What conclusion should you reach?

A:

• K = 6

• F₀.₀₅ ≈ 2.53

• F₀.₀₁ ≈ 3.70

• Since 2.30 < 2.53, p > .05.

• Retain H₀.

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Q: An ANOVA produced F = 5.20 with df = 2, 18.

• How many treatment groups are there?

• What conclusion should you reach?

A:

• K = 3

• F₀.₀₅ ≈ 3.55

• F₀.₀₁ ≈ 6.01

• Since 5.20 > 3.55 but 5.20 < 6.01, p < .05 but p > .01.

• Reject H₀ at .05, but not at .01.

Q: When is a repeated-measures design not appropriate?

When participants cannot reasonably experience every treatment because the treatment is permanent, irreversible, mutually exclusive, or has lasting effects (e.g., surgery, different psychotherapies, vaccines, learning from scratch).

One of the following experiments does not lend itself to a repeated-measures design using the same participants in every treatment. Which one, and why?

a. A researcher wants to compare participants' reaction times after drinking 0 mg, 100 mg, and 200 mg of caffeine, with a one-week washout period between conditions.

b. A researcher wants to compare the effectiveness of three different surgical techniques for repairing torn ACLs by measuring recovery time.

• b does not lend itself to a repeated-measures design.

• Why? A participant cannot undergo three different ACL surgeries on the same injury. The treatment is permanent and cannot be repeated on the same person.

One of the following experiments does not lend itself to a repeated-measures design using the same participants in every treatment. Which one, and why?

a. A psychologist measures participants' stress levels before a mindfulness program, immediately after the program, and one month later.

b. A nutritionist wants to compare the long-term effects of three different weight-loss surgeries on body weight.

• b does not lend itself to a repeated-measures design.

• Why? A participant cannot receive multiple different weight-loss surgeries. The procedures are permanent and mutually exclusive.

One of the following experiments does not lend itself to a repeated-measures design using the same participants in every treatment. Which one, and why?

a. A researcher measures memory performance after participants sleep for 4 hours, 6 hours, and 8 hours, with one week between each condition.

b. A researcher wants to compare three methods of teaching a foreign language from scratch, measuring fluency after a semester.

• b does not lend itself to a repeated-measures design.

• Why? Once participants learn the language using one teaching method, that learning carries over. They cannot "start from scratch" again for the other methods.

One of the following experiments does not lend itself to a repeated-measures design using the same participants in every treatment. Which one, and why?

a. A sports scientist measures sprint speed before a training program, after four weeks, and after eight weeks.

b. A physician wants to compare three different vaccines by measuring whether participants develop immunity after receiving the vaccine.

• b does not lend itself to a repeated-measures design.

• Why? Once a participant receives a vaccine, it affects the immune system. They cannot return to an untreated state to receive the other vaccines independently.

One of the following experiments does not lend itself to a repeated-measures design using the same participants in every treatment. Which one, and why?

a. A researcher measures blood pressure after participants consume 0 g, 5 g, and 10 g of sodium, with several days between each testing session.

b. A researcher wants to compare the effects of three smoking-cessation programs by measuring whether participants remain smoke-free six months later.

Answer:

• b does not lend itself to a repeated-measures design.

• Why? Completing one smoking-cessation program changes participants' behavior and experience, making it impossible to fairly test the other programs as if they were starting fresh.

A repeated-measures design works well when participants can experience every treatment _______ (e.g., caffeine doses, meditation, exercise levels, sleep duration).

independently - It does not work well when treatments are permanent, irreversible, or produce lasting carryover effects (e.g., surgeries, vaccines, psychotherapy, learning a new skill from scratch).