T4: Independent Groups ANOVA (PSYC2010)

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Questions and Answers

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.

False (B)

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?

<p>Perform follow-up tests such as planned comparisons or post hoc tests. (A)</p> Signup and view all the answers

Running multiple t-tests instead of an ANOVA can decrease the Type I error rate.

<p>False (B)</p> Signup and view all the answers

In the context of ANOVA, what does the F-statistic represent?

<p>A proportion of the treatment effect over the differences one might expect due to sampling error. (A)</p> Signup and view all the answers

According to the material, what is the relationship between the F-statistic and the t-statistic when comparing two levels of an independent variable?

<p>the F-statistic is equal to the t-statistic squared (t²)</p> Signup and view all the answers

In a one-way independent-groups ANOVA, the total variance is partitioned into three sources: treatment, error, and random variance.

<p>False (B)</p> Signup and view all the answers

What do $MS_{treatment}$ and $MS_{error}$ represent in ANOVA?

<p>The 'Mean Square' for treatment and error, akin to variance, calculated by dividing the sum of squares by the degrees of freedom. (D)</p> Signup and view all the answers

The formula for calculating the F-statistic in ANOVA is F = $MS_{Treatment}$ / ______.

<p>$MS_{Error}$</p> Signup and view all the answers

Match the ANOVA terms with their definitions:

<p>Total variance = The sum of the variance attributed to the treatment and the variance attributed to the error. Treatment variance = Variation that can be attributed to the differences between treatment groups. Error variance = Variation among the scores within each group/condition that cannot be attributed to the independent variable.</p> Signup and view all the answers

In ANOVA, what does 'SS' stand for, and what does it quantify?

<p>Sum of Squares; variability of the data</p> Signup and view all the answers

What does the notation 'N' represent in ANOVA?

<p>The total number of participants across all groups (D)</p> Signup and view all the answers

What does 'k' represent in ANOVA notation?

<p>The number of levels of an independent variable. (D)</p> Signup and view all the answers

In ANOVA, $n_k$ always signifies the total sample size, regardless of the group.

<p>False (B)</p> Signup and view all the answers

In ANOVA notation, $\overline{X_k}$ represents the ______ of the $k^{th}$ group.

<p>mean</p> Signup and view all the answers

In ANOVA, how is the grand mean ($\overline{X}$) defined?

<p>the mean across all groups</p> Signup and view all the answers

How is $SS_{total}$ calculated in a one-way independent-groups ANOVA?

<p>$SS_{total} = \sum(X - \overline{X})^2$ (D)</p> Signup and view all the answers

$SS_{treatment}$ represents variance that cannot be attributed to the independent variable.

<p>False (B)</p> Signup and view all the answers

In the context of ANOVA, the term '$SS_{error}$' refers to

<p>variance among scores within each group/condition that cannot be attributed to the independent variable (A)</p> Signup and view all the answers

With an independent groups ANOVA, variance is partitioned into what two sources?

<p>Treatment and error (D)</p> Signup and view all the answers

What does F represent in the equation $F = \frac{Between}{Within}$ Condition Variance?

<p>F-statistic</p> Signup and view all the answers

A significant F-statistic always directly indicates which specific group means differ significantly from each other.

<p>False (B)</p> Signup and view all the answers

Match the following terms to their corresponding formulas in the context of ANOVA:

<p>$\overline{X}$ = $\sum X / N$ $MS_{Treatment}$ = $\frac{SS_{Treatment}}{df_{Treatment}}$ $SS_{total}$ = $\sum(X - \overline{X})^2$</p> Signup and view all the answers

In ANOVA, the degrees of freedom for treatment ($df_{treatment}$) is calculated as ______.

<p>k-1</p> Signup and view all the answers

In ANOVA, how is the degrees of freedom for error ($df_{error}$) generally calculated?

<p>$N - k$ (D)</p> Signup and view all the answers

The degrees of freedom total ($df_{total}$) is calculated as $N - k - 1$.

<p>False (B)</p> Signup and view all the answers

After computing the F-statistic, what must you compare it to in order to make a statistical decision?

<p>F critical value</p> Signup and view all the answers

In interpreting the results of an ANOVA, which elements should be included in the conclusion? (Select all that apply)

<p>A mention of the dependent variable and the independent variable (including levels). (A), A statement of the test/analysis used. (B), An indication of whether the result was statistically significant or non-significant. (C), Appropriate statistical notation. (D)</p> Signup and view all the answers

Flashcards

What is ANOVA?

Investigates differences between score distributions across multiple IV levels.

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?

A test of all scores and their variance within a dataset.

What does the F-statistic represent?

The F-statistic represents the proportion of treatment effect over expected sampling error.

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What is the aim of the F-statistic?

Compares variance among treatment means to variance within samples.

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Where does variance come from in a one-way ANOVA?

Variance is partitioned into treatment and error.

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What is the Mean Square?

Sum of squares divided by degrees of freedom.

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What does 'N' stand for?

Total number of participants across all groups.

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What does 'k' stand for?

Number of groups being compared.

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What does 'n_k' stand for?

Number of participants in the kth group.

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What does 'X_i' stand for?

The ith raw score.

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What does 'X_k' stand for?

The mean of the kth group.

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What is SS_total?

Total variation among all scores.

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What is SS_treatment?

Variation attributed to the differences between treatment groups.

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What is SS_error?

Variation not attributed to the IV.

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Name the steps in ANOVA

State hypotheses, statistic, calculate values, and fill in the ANOVA table.

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What decision do we make on our null hypothesis?

If significant, reject the null hypothesis.

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What are degrees of freedom?

The number of scores free to vary given certain restrictions.

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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 (XÌ„ = ΣX / N)
  • SStotal = Σ(X − XÌ„)²
  • SStreatment = Σnk (XÌ„k − XÌ„)²
  • SSerror = Σ(Xik − XÌ„k)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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