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
