Module 2 - One-way ANOVA Flashcards
Comparing Two Means
- Review of the t-test
- Independant-samples
- Paired-samples
Comparing Several Means
- Theory behind the one-way between-subjects ANOVA
- Running a one-way between-subjects ANOVA
The t-statistic - Comparing Two Means
The main purpose of a t-test is to test whether two group
means are significantly (or meaningfully) different from one
another
The t-statistic Types
Between-groups
- When there are two experimental conditions and different participants were assigned to each condition
- Otherwise called independent-samples, independent-measures, independent-means
Repeated-measures
- When there are two experimental conditions and the same participants took part in both conditions of the experiment
- Otherwise called paired-samples, dependent-means, matched pairs
The t-statistic Rationale
Two sample means are calculated
- Under the null hypothesis we expect those means to be roughly equal
- We compare the obtained mean difference against the null hypothesis (no difference)
- We use the standard error as a gauge of the random variability expected between sample means
- If the difference between sample means is larger than expected based on the standard error then:
- There is no effect and this difference has occurred by chance
- There is an effect and the means are meaningfully different
The t-statistic - Independent- samples t-test
- Level of measurement (DV interval or ratio)
- Random sampling
- Normality
- Homogeneity of variance
(Sample Means) - (Null Hypothesis [0]) / Average Standard
The t-statistic - Paired- samples t-test
- Level of measurement (DV interval or ratio)
- Random sampling
- Normality
(Mean difference of scores) - (Null Hypothesis [0]) / Average Standard error of Differences
One-Way ANOVA
Comparing several means
The main purpose of a one-way ANOVA is for situations where we want to compare more than two conditions
Type I Error
False Positive
Type II Error
False Negative
Familywise error rate (FWER)
For a single comparison using α=.05 the probability of a type 1 error is 5%
With the addition of another comparison using α=.05
One-way ANOVA test
Null hypothesis
H0: μ1 = μ2 = μ3
Alternative hypothesis
H1: At least 1 group is different from another
One-Way ANOVA
The ANOVA produces an F-statistic or an F-ratio
The F-statistic represents the ratio of the model to its error
ANOVA is an omnibus test
Tests for an overall experimental effect
Significant F-statistic tells us that there is a difference somewhere between the groups but not where this difference lies
F - test
F = Variability between groups / Variability within groups = (Random Error) + (Treatment Effect) / Random Error
If Null is true, treatment effect will be 0
If treatment effect increases, F value increases
Squared Sums Total
All the variance in mood
𝑆𝑆𝑇 =∑ ( 𝑋 − 𝑋g𝑟𝑎𝑛𝑑 ) ²
∑ (sum of)
𝑋 (our values)
𝑋g𝑟𝑎𝑛𝑑 (Grand x-bar [mean])
Squared Sums Between
Variance in mood explained by our model
𝑆𝑆𝐵 =∑ 𝑛𝑖 ( 𝑋𝑖 − 𝑋𝑔𝑟𝑎𝑛𝑑 ) ²
∑ (sum of)
𝑛𝑖 (n [the number] in I [this group])
𝑋𝑖 (mean of this group)
𝑋𝑔𝑟𝑎𝑛𝑑 (Grand mean)
Squared Sums Within
Variance in mood not explained by our mode
𝑆𝑆𝑊 =∑ ( 𝑋 − 𝑋𝑔𝑟𝑜𝑢𝑝 )
∑ (sum of)
𝑋 (raw values)
𝑋𝑔𝑟𝑜𝑢𝑝 (group means)
Degrees of freedom
SStotal - Entire sample -1
SSbetween - Group means -1
SSwithin - Entire sample -Number of groups (n-k)
One-Way ANOVA Assumptions - Level of measurement
Dependent variable must be measured at the interval or ratio level
One-Way ANOVA Assumptions - Random sampling
Scores must be obtained using a random sample from the population of interest
One-Way ANOVA Assumptions - Independence of observations
The observations that make up the data must be independent of one another
Violation of this assumption is very serious as it dramatically increases the Type 1 error rate
One-Way ANOVA Assumptions - Normal distribution
The populations from which the sample are taken is assumed to be normally distributed
Need to check this for each group separately in one-way ANOVA
One-Way ANOVA Assumptions - Homogeneity of variance
Samples are obtained from populations of equal variances
ANOVA is fairly robust to this violation – provided the size of your groups are reasonably similar
One-Way ANOVA Assumptions
Level of measurement – met (or not) by design
Random sampling – met (or not) by design
Independence of observations – met (or not) by design
Checking assumptions in SPSS
Reporting One-Way ANOVA - Reporting assumptions - Normality
The Kolmogorov-Smirnov test revealed that mood scores were not normally distributed for the low intensity exercise group, D (5) = .35, p = .046, or the high intensity exercise group, D (5) = .03, p = .026. Mood was normally distributed in the medium intensity exercise group, D (5) = .23, p = .20. This data indicates that the
assumption of normality was violated. However, ANOVA is robust to violations of normality when the sample size is large and there are roughly equal numbers of participants across experimental groups, therefore this violation was ignored (Field, 2007).
Reporting One-Way ANOVA - Reporting assumptions - Homogeneity of Variance
The variances for mood across experimental groups were roughly equal as indicated by the non-significant Levene statistic, F (2, 12) = .64, p = .55. The assumption of homogeneity of variances was not violated
Reporting ANOVA
A one-way ANOVA test revealed that there was a significant effect of exercise group on mood scores F (2,12) = 5.12, p = .025
Between & Within group DF
F statistic
Unless p is less than .001 in which case we report p < .001
Module Summary
ANOVAs can be used to compare multiple conditions
You analyse the variance between experimental conditions and compare that to the variance within a condition (random error)
F-ratios increase in magnitude as a treatment effect increases
There are a number of assumptions that we need to check, and know how to report
