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Questions and Answers
What is the primary purpose of conducting an ANOVA?
What is the primary purpose of conducting an ANOVA?
- To compare the means of two groups.
- To determine the correlation between two continuous variables.
- To predict the value of a dependent variable based on an independent variable.
- To investigate differences between distributions of scores across two or more levels of two or more IVs. (correct)
A one-way ANOVA can only be used when the independent variable has exactly two levels.
A one-way ANOVA can only be used when the independent variable has exactly two levels.
False (B)
What does an omnibus test in ANOVA determine?
What does an omnibus test in ANOVA determine?
- The interaction effects between independent variables.
- The specific pairs of means that are significantly different.
- The effect size of each individual independent variable.
- Whether there is a significant difference between any of the means. (correct)
If an ANOVA yields a significant omnibus test, what is the next step to determine which groups differ significantly?
If an ANOVA yields a significant omnibus test, what is the next step to determine which groups differ significantly?
Running multiple t-tests instead of an ANOVA can decrease the Type I error rate.
Running multiple t-tests instead of an ANOVA can decrease the Type I error rate.
In the context of ANOVA, what does the F-statistic represent?
In the context of ANOVA, what does the F-statistic represent?
According to the material, what is the relationship between the F-statistic and the t-statistic when comparing two levels of an independent variable?
According to the material, what is the relationship between the F-statistic and the t-statistic when comparing two levels of an independent variable?
In a one-way independent-groups ANOVA, the total variance is partitioned into three sources: treatment, error, and random variance.
In a one-way independent-groups ANOVA, the total variance is partitioned into three sources: treatment, error, and random variance.
What do $MS_{treatment}$ and $MS_{error}$ represent in ANOVA?
What do $MS_{treatment}$ and $MS_{error}$ represent in ANOVA?
The formula for calculating the F-statistic in ANOVA is F = $MS_{Treatment}$ / ______.
The formula for calculating the F-statistic in ANOVA is F = $MS_{Treatment}$ / ______.
Match the ANOVA terms with their definitions:
Match the ANOVA terms with their definitions:
In ANOVA, what does 'SS' stand for, and what does it quantify?
In ANOVA, what does 'SS' stand for, and what does it quantify?
What does the notation 'N' represent in ANOVA?
What does the notation 'N' represent in ANOVA?
What does 'k' represent in ANOVA notation?
What does 'k' represent in ANOVA notation?
In ANOVA, $n_k$ always signifies the total sample size, regardless of the group.
In ANOVA, $n_k$ always signifies the total sample size, regardless of the group.
In ANOVA notation, $\overline{X_k}$ represents the ______ of the $k^{th}$ group.
In ANOVA notation, $\overline{X_k}$ represents the ______ of the $k^{th}$ group.
In ANOVA, how is the grand mean ($\overline{X}$) defined?
In ANOVA, how is the grand mean ($\overline{X}$) defined?
How is $SS_{total}$ calculated in a one-way independent-groups ANOVA?
How is $SS_{total}$ calculated in a one-way independent-groups ANOVA?
$SS_{treatment}$ represents variance that cannot be attributed to the independent variable.
$SS_{treatment}$ represents variance that cannot be attributed to the independent variable.
In the context of ANOVA, the term '$SS_{error}$' refers to
In the context of ANOVA, the term '$SS_{error}$' refers to
With an independent groups ANOVA, variance is partitioned into what two sources?
With an independent groups ANOVA, variance is partitioned into what two sources?
What does F represent in the equation $F = \frac{Between}{Within}$ Condition Variance?
What does F represent in the equation $F = \frac{Between}{Within}$ Condition Variance?
A significant F-statistic always directly indicates which specific group means differ significantly from each other.
A significant F-statistic always directly indicates which specific group means differ significantly from each other.
Match the following terms to their corresponding formulas in the context of ANOVA:
Match the following terms to their corresponding formulas in the context of ANOVA:
In ANOVA, the degrees of freedom for treatment ($df_{treatment}$) is calculated as ______.
In ANOVA, the degrees of freedom for treatment ($df_{treatment}$) is calculated as ______.
In ANOVA, how is the degrees of freedom for error ($df_{error}$) generally calculated?
In ANOVA, how is the degrees of freedom for error ($df_{error}$) generally calculated?
The degrees of freedom total ($df_{total}$) is calculated as $N - k - 1$.
The degrees of freedom total ($df_{total}$) is calculated as $N - k - 1$.
After computing the F-statistic, what must you compare it to in order to make a statistical decision?
After computing the F-statistic, what must you compare it to in order to make a statistical decision?
In interpreting the results of an ANOVA, which elements should be included in the conclusion? (Select all that apply)
In interpreting the results of an ANOVA, which elements should be included in the conclusion? (Select all that apply)
Flashcards
What is ANOVA?
What is ANOVA?
Investigates differences between score distributions across multiple IV levels.
What is a one-way ANOVA?
What is a one-way ANOVA?
Focuses on one IV's effect on the DV. Requires the IV has more than two levels.
What is an omnibus test?
What is an omnibus test?
A test of all scores and their variance within a dataset.
What does the F-statistic represent?
What does the F-statistic represent?
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What is the aim of the F-statistic?
What is the aim of the F-statistic?
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Where does variance come from in a one-way ANOVA?
Where does variance come from in a one-way ANOVA?
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What is the Mean Square?
What is the Mean Square?
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What does 'N' stand for?
What does 'N' stand for?
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What does 'k' stand for?
What does 'k' stand for?
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What does 'n_k' stand for?
What does 'n_k' stand for?
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What does 'X_i' stand for?
What does 'X_i' stand for?
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What does 'X_k' stand for?
What does 'X_k' stand for?
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What is SS_total?
What is SS_total?
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What is SS_treatment?
What is SS_treatment?
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What is SS_error?
What is SS_error?
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Name the steps in ANOVA
Name the steps in ANOVA
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What decision do we make on our null hypothesis?
What decision do we make on our null hypothesis?
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What are degrees of freedom?
What are degrees of freedom?
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Study Notes
- ANOVA assignment is due May 6th
- This assignment makes up 35% of the final grade
- Material from Tutorials 4 and 5 is covered in the assignment
- ANOVA can investigate the differences between distributions of scores across two or more levels of two or more Independent Variables (IVs)
- A one-way ANOVA focuses on the effect of one IV on the Dependent Variable (DV)
- IVs can have more than two levels, unlike a t-test
- ANOVA provides an omnibus test, a test of all scores and their variance within a dataset
- A significant omnibus test indicates a difference exists between means, but follow-up tests must establish where the differences exist
One-Way ANOVA Example
- IV = Drug Dosage (None, Low, Medium, High)
- DV = rats' maze running times
- T-tests can only compare two groups of scores (e.g., Low vs Medium)
- Many t-tests would have to run to investigate all possible group combinations, which would inflate the Type I error rate
- Instead, one F-test is run, and only attempts to find where any differences exist if the omnibus test is significant
- If F is significant, perform planned or post hoc comparisons (typically corrected t-tests) to find out where exactly differences exist
Understanding the F-Statistic
- The F-statistic, like t, represents a proportion of the treatment effect over differences due to sampling error
- For independent groups: t = (X1 - X2) / sX1-X2
- F = Treatment / Error
- Aims to compare the variance among the treatment means to the variance within the samples themselves
Partitioning Variance in One-Way ANOVA
- Variance is partitioned into two sources: treatment and error
- F = MSTreatment / MSError
- MStreatment and MSerror is a "Mean Square", like variance: sum of squares (SS) divided by the degrees of freedom (df)
- MStreatment = SSTreatment / dfTreatment
- MSError = SSError / dfError
- F = Between / Within Condition Variance
Notation
- N = Total number of participants (across all groups)
- k = Number of groups
- nk = Number of participants in the kth group
- Xi = The ith raw score (also X)
- Xik = The ith raw score in the kth group
- Xk = The mean of the kth group
- X. = The grand mean (across all groups)
- SStotal = SStreatment + SSerror
Sum of Squares Types
- SStotal = Σ(X – X.)²: total variation among all scores
- SStreatment = Σnk (Xk – X.)²: variation attributable to differences between treatment groups (i.e., at each IV level)
- SSerror = Σ(Xik – Xk)²: variation among the scores within each group/condition that cannot be attributed to the IV (i.e., residual)
Conducting an Independent Groups ANOVA
- State the variables and hypotheses
- State the statistic
- Calculate
- Grand mean (X̄ = ΣX / N)
- SStotal = Σ(X − X̄)²
- SStreatment = Σnk (X̄k − X̄)²
- SSerror = Σ(Xik − X̄k)²
- MStreatment and MSerror (MS = SS / df)
- F = MStreatment / MSerror
- Fill in the ANOVA summary table
- Compare Fobt to Fcrit using an F-table, rejecting H0 if Fobt > Fcrit
- Interpretation and conclusion should include
- A test/analysis statement
- Significance indication (significant/non-significant)
- Mention of the DV and IV (and all IV levels)
- Statistical notation (F(dftreatment, dferror) =, p < .05 or ns)
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