statistical analysis and design Flashcards
1
Q
what is correlational research?
A
= non-experimental study which determines the relationship between two variables without manipulating them or controlling for extraneous variables
2
Q
why would we use correlational studies?
A
- when it would be unethical and harmful to manipulate variables
- allows the researcher to observe natural variations
3
Q
what is a key limitation of correlational studies?
A
- ## correlation does not equal causation
- the third variable problem -> two variables can be statistically related but not because they cause each other but because a third variable causes both of them.
4
Q
what are the two characteristics of a correlational relationship
A
- the direction:
positive correlation= both variables increase together
negative correlation= one variable increases and the other decreases - the strength
- measured using covariance and correlation coefficient (r)
- positive covariance= both variables tend to increase or decrease together
- negative covariance= when one variable is high, the other tends to be low
5
Q
how to calculate the sample covariance of two variables
A
covariance = a measure of how the two variables change together.
limitations= affected by units of measurement - indicates direction but not strength
step 1= for each participant, subtract their value from the mean, for both variables, and multiply them together, giving one value for each participant
step 2= sum all the values created in step 1 together
step 3= divide by the number of pairs of observations minus 1
if covariance is + -> positive relationship
if covariance is - -> negative relationship
6
Q
how do you standardise covariance across different units of measurement and what is the formula for this?
(makes the covariance easier to compare between different pairs of varia
A
= transform covariance into a correlation coefficient (r)
- indicates strength and direction
Range: -1 to +1
Interpretation:
|r| ≈ 0.1-0.3 → Weak
|r| ≈ 0.4-0.6 → Moderate
|r| ≈ 0.7-0.9 → Strong
1 = perfect
calculated by:
the covariance calculated/ the SD of variable 1 multiplied by the SD for variable 2
7
Q
null hypothesis significance testing in correlational analysis
A
Null Hypothesis (H₀): No correlation (r = 0).
Alternative Hypothesis (H₁): Significant correlation (r ≠ 0).
- when stating a directional hypothesis, only state direction, not the strength (strong, moderate, weak)
7
Q
A
8
Q
what are the assumptions of a Pearson correlation as a parametric test?
A
- levels of measurement:
- interval or ratio (continuous) variables - related pairs
- each observation has two paired values - linearity
- relationship between variables should be linear
- Check with residuals vs. fitted plot (flat red line = linear). - normality
- residuals are normally distributed
- Check with Q-Q plot (values close to diagonal). - homoscedasticity
- the variability or spread of one variable remains constant across the range of another variable
- Check with scale-location plot (flat red line). - absence of outliers
- can distort results
9
Q
what can we use to visualise relationship between two continuous variables?
A
a scatterplot
10
Q
how to calculate degrees of freedom for Pearson correlation
A
df = N-2
- one degree of freedom for each variable
11
Q
Spearman correlation as a non parametric test: what is it and when do we use it?
A
= calculates the relationship based on the rank order of the data, rather than the actual values.
we use when:
- the data is ordinal (ranked 1,2,3,4,5)
- the data violates assumptions of Pearson correlation
- relationship between variables is non-linear
- the residuals are not normally distributed
12
Q
what are the steps to calculate the spearman correlation coefficient (rs)
A
- rank the scores for each variable separately
- Smallest value gets rank 1, next smallest gets 2, etc. - calculate the differences between ranked pairs (not the original scores) = d
-( variable 1 ranked score - variable 2 ranked score) - square the differences = d2
- sum/ add the squared differences
- plug values into Spearman’s formula, where n = number of observations
- used when there are no tied ranks
12
Q
A
13
Q
how to handle tied observations Spearman correlation datasets
A
- when two or more observations have the same value, we have ties in the data(difficult to rank) -> need to calculate tied rank
- identify what ranks that a tie of two observations shoud get naturally
- assign the average of the tied ranks
for example: If two values would be ranked 3 and 4, assign both 3.5.
if more than two values, add all natural ranks together and divide by how mnay tied ranks there are
- do this for all observation sets that are tied
Tied ranks slightly affect standard Spearman’s formula.
Alternative approach: Apply Pearson correlation on the rank scores.
14
Q
compare independent and dependent correlation coefficients
A
- independent correlations = Two correlations come from different, unrelated groups
eg -> Comparing correlation between TikTok usage & GPA for high school vs. college students. - dependent correlations= Two correlations share a common variable.
eg -> Comparing correlation between study hours & statistical anxiety vs. study hours & attitude towards statistics (common variable: study hours).
15
Q
describe hypothesis testing for correlation coefficients
A
Null Hypothesis (H₀): No difference between the two correlations.
Alternative Hypothesis (H₁): A statistically significant difference exists.
Use ‘cocor’ package in R to compare correlation coefficients.
16
Q
the r critical value
A
= is the smallest r-value you need to find a significant effect
- if the r value that yiu have calculated is equal to or greater than the r critical value at a particular significance level, then your test is significant
- don’t take into account polarity (-/+)
17
Q
the difference between strength and direction in a relationship
A
Direction= Indicates whether the relationship is positive or negative.
Strength = Indicates how closely the two variables follow a linear pattern.
Measured by correlation coefficient (r)
A strong correlation means data points are close to a straight line, while a weak correlation means they are more scattered.
example
r = 0.9 -> direction = positive, strength = strong
18
Q
cohens rule of thumb for effect size
A
small = 0.1
medium = 0.3
large = 0.5
- not the same as correlation coefficient values where 0.5 would be considered moderate and 1 as perfect (-1-1)
19
Q
what is the difference between correlation and regression studies
A
Correlation: Measures the strength & direction of the relationship between two variables.
Example: Is ease of purchase correlated with purchase intention?
Regression: Determines whether one variable predicts another.
Example: Does ease of purchase predict purchase intention?
Key distinction: Correlation does not imply causation, while regression models predictive relationships.
20
Q
describe the variables in regression analysis
A
predictor variable:
- the independent variable
- the explanatory variable
- the x variable in the regression model
- eg each of purchase
outcome variable:
- the dependent variable
- the criterion variable
- the y variable in the regression model
- eg purchase intention
21
Q
what is regression analysis
A
= A statistical technique that models relationships between variables.
Answers: “By how much will Y change if X changes?”
Types:
Simple Linear Regression: One predictor (X) prediciting → One outcome (Y).
Example: Does ease of purchase predict purchase intention?
Multiple Linear Regression: Two or more predictors (X1, X2, etc.) predicting→ One outcome (Y).
Example: Do ease of purchase and influencer endorsements predict purchase intention?
22
Q
the mean model
used to make predictions in regression
A
mean model:
Predicts the outcome (Y) using the mean of all responses.
Ignores the predictor variable.
Example:
- Model predicts that the outcome variable(purchase intention) will always be the mean value, independent of the predictor variable value (ease of purchase)
- however the actual purchase intention is often higher or lower than the mean value
- this model is not good as it does not capture the data well
23
the regression model
| better option over the mean model
- Uses the best-fit line (regression line) to adjust predictions based on predictor variables.
- More accurate than the mean model.
- the prediction of purchase intention changes based on the value of ease of purchase
- not a perfect prediction for every point
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the regression line
| line of best fit
-line represents the predicted values.
-May pass through some, all, or none of the actual points.
25
residuals in regression
| error
Residual = difference between what the model predicted and the actual value
-> Actual Value – Predicted Value
- Measures prediction accuracy. (how much error is in the predictions)
-Smaller residuals = better model fit. (lower error)
- larger residuals = worse predictions/ higher error
-Positive residuals -> when actual value is above regression line -> model underestimated outcome
eg the actual purchase intention is higher than predicted
-Negative residuals -> when actual value is below regression line -> model overestimated outcome.
eg the actual purchase intention is lower than predicted
26
the simple linear regression equation
Y = b0 + b1X + error
-Y hat = the predicted outcome based on the regression model
- X = the predictor variable (the factor influencing Y)
- b0 (beta-zero) = the intercept, representing the predicted value of Y when X = 0
- b1 (beta-one) = represents the change in the predicted Y for each 1 unit increase of X (the slope)
- error = differences between actual and predicted values (measure of residuals)
Example: If b₁ = 4.41, then a 1-hour increase in study time leads to a 4.41-point increase in exam score.
27
how to calculate the slope and intercept in simple linear regression equation
| b1 and b0
b0 = y - b1 * x
b1 = r * (sy/sx)
x and y are the mean values of x and y
r is the pearson correlation coefficient
sx and sy are the standard deviations of X and Y
28
breaking down the simple linear regression equation with an example
imagine we are studying the effect of study hours on exam scores.
Let's say we find the equation:
Exam Score = 60 + 4X
Here’s what this means:
Intercept (60) → If a student studies 0 hours, they are predicted to score 60 on the exam.
Slope (4) → For each additional hour of study, the exam score increases by 4 points.
Example Calculations
If a student studies 2 hours, their predicted score is: 60 + (4 x 2) = 68
if a student studies 5 hours, their predicted score is:
60 + (4 x 5) = 80
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key takeaways of the simple linear regression equation
Y (Predicted Outcome) → This is what we are trying to predict.
b0 (intercept) -> This is the value of Y when X=0. It represents the starting point of our line.
b1 (slope) -> this tells us how much y changes when x increases by 1 unit.
x (predictor variable) -> this is the variable that helps predict y
30
the basic algebraic formula in the linear regression model
y = a + bx
y = outcome variable
x = predictor variable
b = slope (how much y changes when x increases by 1 unit)
eg if the slope b = 3, then for every 1-unit increase in ease of purchase (x), purchase intention (y) increases by 3 points
31
what are the assumptions of a simple linear regression
1. Levels of Measurement:
Outcome (DV): Continuous (interval/ratio).
Predictor (IV): Continuous or categorical (with 2 levels).
2. Independence: Each observation/score must come from a different participant.
3. Non-Zero Variance: Predictor values must vary (spread in the data). - can check using the scatterplot to ensure the predictor values are not all the same
4. Linearity: The relationship between predictor and outcome must be linear. Checked via Residuals vs. Fitted Plot with residuals on y-axis and predicted values on x-axis (flat red line).
5. Normality: Residuals should follow a normal distribution. Checked via Q-Q Plot with standardised residuals from the model- majority of points lying on dashed line and Shapiro-Wilk test (p > 0.05 = assumption met, p<0.05- violated).
6. Homoscedasticity: The spread of residuals is constant along the values of the outcome variable. Checked via Scale-Location Plot (flat red line) which shows the square root of the standardised residuals on y and fitted values on x and Breusch-Pagan test (p > 0.05 = assumption met, p<0.05 - violated, heteroscedasticity).
32
what are the 3 key concepts for analysing and interpreting simple linear regression
1. r-squared
2. f- value
3. regression co-efficient
33
r squared in a SLR
R² (Coefficient of Determination):
= Proportion of variance in the outcome variable that is explained by the predictor variable.
- values vary between 0 and 1 with 1 being the perfect prediction of the outcome variable.
Example: R² = 0.34 → 34% of the variance explained by the model
- 66% unexplained and may be due to other factors not included in the model.
- the r2 itself does not tell us if this explanation is statistically significant.
Adjusted R² compensates for model complexity.
34
the f-test and f value in SLR
f value = determines if the model explains a statistically significant amount of variance.
Example: F(1,98) = 49.45, p < .001 → Model is significant.
f-test (ANOVA- analysis of variance) to evaluate if R2 is statistically significant.
R² tells us how much is explained, and the F-test tells us whether what is explained is statistically significant
35
the regression coefficient b in SLR
= Indicates the direction & strength of the relationship.
Example: b = 0.31, p < .001 → A 1-unit increase in Cognitive Cultural Intelligence increases Psychological Adaptation by 0.31 units.
positive value = the predictor and outcome variable move in the same direction
eg (b) = 2.5 -> For every additional hour of study, grades improve by 2.5 points
negative value = the predictor and outcome variable move in opposite directions
(b) = -0.8 -> For every additional hour of study hours, grades decrease by 0.8 points
- value found under estimate heading for predictor variable in r code
36
Effect Size (Cohen’s f² in SLR
Small = 0.02, Medium = 0.15, Large = 0.35
Formula: f² = (R² adj / (1 - R² adj))
(R2 adjusted divided by 1 minus R2 adjusted)
Example: f² = 0.49 (Large effect).
37
adjusted R- squared in SLR
- adjusts the R2 for the number of predictors there are in the model so we report it when there is more than one predictor
- it provides a more accurate measure of the goodness of fit - tells us more about the model accuracy when more than one predictor
- value is always less or equal to R2
38
degrees of freedom in a SLR
Regression df = 1 (for one predictor, slope).
Residual df = n - 2 (adjusted for intercept & slope).
report both values in the write up- (1,98)
39
what is multi linear regression
=Two or more predictor variables → One outcome variable
- allows us to analyze the unique contribution of multiple predictors while controlling for others.
40
give an example of a multi linear regression study
Outcome Variable: Academic Engagement
Predictors:
Positive emotions (continuous)
Psychological capital (continuous)
Student connectedness (continuous)
Extracurricular participation (categorical, two levels)
41
what is the multi linear regression equation
Y = b0 +b1X1 + b2X2 + b3X3 + b4X4 + error
y = outcome variable (dependent)
X1, X2, X3 X4 = the predictor variables
b0 = the intercept, the predicted value of y when all predictors are 0, baseline level
b1, b2, b3, b4 = the change in the predicted Y for each one-unit increase in each predictor
error = measure of residuals
- the number of x values you have is proportionate to the number of predictor variables you have
- All other variables are held constant when interpreting each coefficient.
- multiplying b values by the corrosponding x
42
example using the multiple linear regression equation
suppose we are predicting academic engagement (Y) based on the 4 predicor values
the equation = Academic Engagement=2.1+0.45(Pos. Emotions)+0.30(Psych. Capital)+0.25(Student Conn.)+1.2(Extracurricular Participation)+ϵ
Interpretation of Coefficients:
Intercept (2.1): When all predictors are 0, predicted academic engagement is 2.1.
0.45 (Pos. Emotions): For each 1-unit increase in Positive Emotions, Academic Engagement increases by 0.45, holding other variables constant.
1.2 (Extracurricular Participation): Since this is categorical (e.g., 0 = No, 1 = Yes), students who participate in extracurriculars score 1.2 units higher in engagement compared to those who don't.
43
the assumptions of a MLR
| same as simple with one additon
1. levels of measurement (outcome variable is interval/ratio data, predictors are interval/ratio or categorical with two levels)
2. independence
3. non-zero variance
4. linearity
5. nomality of residuals
6. homoscedasticity
7. no multicollinearity (predictors should not be highly correlated)
44
tests to check the assumptions of MLR
Linearity: Residuals should be randomly scattered around zero (red trend line ≈ horizontal).
Normality: Use Q-Q plot or Shapiro-Wilk test (p > 0.05 = normally distributed residuals).
Homoscedasticity: Scale-location plot should be flat, no funnel pattern.
Multicollinearity:
Correlation between predictors should be < 0.8.
Variance Inflation Factor (VIF) should be < 5.
45
the no multicollinearity assumption of a MLR
-When predictor variables are highly correlated they overlap in explained variance. (think r coefficient)
This makes it difficult to determine the unique contribution of each predictor
- we don't know which predictor is contributing to the outcome variable
- a little bit of multicollinearity is not a problem but when correlation between variables is strong this is a problem if we want to estimate effects of individual variables
Detecting Multicollinearity:
1. high correlation coefficients - *correlation matrix* (if the r value between two predictors is greater than .80 then they are measuring something similar bringing overlapping information) <0.8, assumption is met
2. Variance Inflation Factor (VIF) (all values should be < 5 to meet assumption),
-values of 5 of larger indicate large multicollinearity problem
46
how to calculate a variance inflation factor by hand
| used to detect the presence of multicollinearity
= 1/ (1-Ri2)
- Ri2 is the R2 value obtained by regressing the i predictor on the remaining predictor variables
- we need to run a linear regression model for each predictor
47
what are the key concepts for analysing and interpreting the results of a MLR
1. R-squared (R²): Proportion of variance explained by the model.
2. Adjusted R-squared: Adjusted for number of predictors.
3. F-value: Tests overall model significance.
4. Regression Coefficient (B): Shows the effect of each predictor on the outcome.
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an example r interpretation for a MLR output
Adjusted R² = 0.51: The model explains 51.3% of variance in academic engagement (outcome)
F(4, 295) = 79.74, p < .001 → Model is statistically significant.
Degrees of Freedom (4, 295 = n - number of variables (4 variables, n = 300 → df = 295).
b coeffiecient for one of the predictors = 5.01
eg -> positive emotions positively predict academic engagement where for every one unit increase in positive emotions, we expect academic engagement to increase by 5.01 (b = 5.01, p< .001)
- value found under estimate std
49
how can we interpret a predictor variable with a categorical level
- look at p and b values
- we can also look at mean values, descriptives and visualise using a violin plot ( eg yes vs no as the two levels)
50
Standardized vs. Unstandardized Regression Coefficients in MLR
Unstandardized Coefficients (b): Expressed in the original units of each predictor. (from the R output) - each predictor is not measured in the same units
Standardized Coefficients (β): Converted to a common scale → allows for direct comparison of predictor importance and can interpret whether residuals are large or not
- we do this by using the scale() function on R
Higher standardized B = More important predictor.
Categorical predictors (2 levels) do not need standardization.
- Can report unstandardised values, however we need to standardised if we want to figure out which predictor is the strongest - making comparisons between them
51
calculating effect size in MLR
Effect Size (D): Higher B = stronger effect.
Example: D = 1.04 → Very large effect.
f2= R2 adjusted/ (1-R2 adjusted)
small = 0.02
medium = 0.15
large = 0.35
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how to write up the stats in a MLR for a report
F (dfs) = f value, p value, adjusted r2 value, effect size, accounting for ...% of the variance.
54
describe the t-test
=A parametric test used to examine whether the difference between two means is statistically significant.
Independent t-test: Used when comparing two independent groups.
- students t test assumes equal variance
- welch's t test does not assume equal variance
- if we have more than 2 we cannot use this method
Non-parametric alternative: Mann-Whitney U Test (when assumptions are not met).
If comparing more than two groups, multiple t-tests are required, leading to an increased risk of Type I error.
55
the problem with conducitng multiple t-tests
Comparing four vocabulary learning methods requires six independent t-tests:
L1 direct translation vs. L2 definitions
L1 direct translation vs. Loci method
L1 direct translation vs. Reminiscence
L2 definitions vs. Loci method
L2 definitions vs. Reminiscence
Loci method vs. Reminiscence
Each test has a 5% chance (α = 0.05) of a Type I error.
Conducting multiple tests accumulates error:
Cumulative probability of making at least one Type I error in 6 tests = 0.26 (26%), instead of 5%.
1- (1-0.05) to the power of 6 = 0.26
6 is the number of tests ran
Capitalizing on chance: More tests → higher chance of false positives.
Solution: ANOVA (Analysis of Variance) instead of multiple t-tests.
56
comparing type 1 and type 2 errors
if the null hypothesis is true but the researcher rejects it and says it is false = type 1 error (false positive)
if the null hypothesis is false but the researcher says it is true and fails to reject it = type 2 error (false negative)
57
capitalising on chance
= conducting so many tests with 0.05 a level on the same data, resulting in the increased likelihood of a type 1 error
More t-tests -> more chance of capitalising on chance -> more chance of type 1 error
- use ANOVA instead
58
what is ANOVA
ANOVA (Analysis of Variance) is a parametric inferential statistical test used to compare the means of three or more groups at once.
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60
one way between subjects ANOVA
-One independent variable (IV).
-Each participant is in only one condition/level (randomly assigned).
61
what are the fundamentals of one way between subjects ANOVA
1. between group variance
2. within group variance
3. f-ratio
ANOVA breaks total variation into between-group variance and within-group variance.
Variance defines and measures the differences among the means of three or more groups
62
between group variance in ANOVA and where does it arise from
= The variability in data due to differences between the groups.
When the means of the groups are different, this indicates that there is a greater degree of variation between the conditions/levels.
If no differences in the means of the groups are found, this indicates that there is no variation.
Sources:
1. Treatment Effects: Different interventions (treatments/conditions) lead to differences in performance.-> effect of the different treatments
2. Individual Differences: Participants naturally vary in the way they respond to tasks
3. Experimental Error: Measurement inconsistencies, procedural issues, uncontrolled external factors, randomness
63
within group variance
= Variability within each group.
Sources:
1. Individual Differences: Even within the same group, participants have different abilities, knowledge or personality traits
2. Experimental Error: Uncontrolled random factors affecting results, randomness in measurement, errors, procedural inconsistencies
64
key differences between within abd between group variance in ANOVA
between- measures the combined effect of error and treatment
within- only measures the effect of error
- all participants within any given group receive the same treatment, so the variance in the group cannot be due to this, variance is only due to random error.
65
partitioning variance in ANOVA
= breaks the overall variation observed in the data (total) into components
steps:
1. calculate the mean for each grouo
2. calculate grand mean -> add up all individual scores from all groups and then divide by total number of observations
3. Calculate within-groups variance: Differences of individual scores from group means.
4. Calculate between-groups variance: Differences between group means and the grand mean.
66
the f-statistic (f ratio) in ANOVA
= the test statistic for ANOVA
- values can vary from 0 to infinity
f = between groups variance/ within groups variance
- this is equal to treatment effect + error/ error
Interpretation:
Higher F-ratio = greater between group variance -> greater difference among group means that are more likely not due to chance.
If F is significant, we reject H₀ (null hypothesis: no difference between groups).
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effect size in ANOVA
Partial Eta Squared (ηₚ²):
0.01 = Small effect
0.06 = Medium effect
0.14 = Large effect
Example: ηₚ² = 0.02 → Small effect size.
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one way between subjects ANOVA
= Used when there is one independent variable (IV) with three or more levels, and each participant appears in only one condition.
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what are the 4 assumptions of the one way between subjects ANOVA
1. Levels of Measurement: The dependent variable (DV) must be continuous (interval or ratio data).
2. Independence: Each observation must be independent (each participant provides one score).
3. Normality: Residuals should be normally distributed.
Check: Shapiro-Wilk test (p > 0.05 = assumption met)
Q-Q residuals plot - majority of points falling alomg dashed line
4. Homogeneity of Variance: The variance around the mean should be equal across groups.
Check: Levene’s Test (p > 0.05 = assumption met)
Graphical check: Boxplots and residuals vs. fits plot.- look for flat line
- large variance makes it harder to detect true differences
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what should we do if assumptions for a one way between subjects ANOVA are violated?
1. If sample sizes are equal between groups & effect sizes are large → ANOVA can still be used.
2. If variance assumption is violated - (heterogeneity) → Use Welch’s ANOVA (does not assume equal variance), however residual assumption must be met to do this.
3. If normality is violated → Use Kruskal-Wallis test (non-parametric alternative).
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Interpreting one way between subject ANOVA Results
F-statistic: Determines if at least one group mean is significantly different.
p-value: If p < 0.05, there is a significant difference.
Effect size: Partial eta squared (ηp²).
ηp² = 0.01 (small), 0.06 (medium), 0.14+ (large).
df1 = number of groups - 1
df2 = number of observations - number of groups (3, 96)
Example Write-Up: A one-way ANOVA showed a significant effect of learning method on vocabulary retention, F(3, 96) = 10.37, p < .001, ηp² = .24.
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post hoc tests for one way between subject ANOVA
Needed if ANOVA results are significant to determine which groups differ.
- which groups have significant differences between them
- maintains a level 0.05
Common post-hoc tests:
1. Tukey's HSD: Compares all group pairs while controlling for Type I error. (looking at individual p values for each comparison)
2. Bonferroni: More conservative, adjusts α-level for multiple comparisons calculates cohen’s d effect size for each pairwise comparisons.
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Key Differences Between One-Way Repeated Measures and Between-Subjects Designs
Repeated Measures ANOVA (within subjects): Each participant experiences all levels of the independent variable.
Between-Subjects ANOVA: Each participant is exposed to only one level of the independent variable.
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benefits and costs of one way between subjects ANOVA
benefits:
simplicity
costs:
- large variation from person to person
- requires large sample sizes for power
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benefits and costs of one way repeated measures ANOVA
benefits:
- more economical - fewer people needed
- making contrasts within each participant
- providing relatively precise estimates- acurately detecting the effect of the conditions or treatments being tested
costs:
- carryover effects
- practise effects
- fatigue effects
- exposure to treatment at one time influences responses at another
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assumptions of a one-way repeated measures ANOVA
1. Levels of Measurement
The dependent variable must be continuous (interval or ratio data).
2. Normality
Residuals should be normally distributed.
How to check:
Q-Q plot: Points should follow a straight line.
Shapiro-Wilk test: Each comparison should have p > 0.05.
3. Sphericity
= the variance of the differences between all pairs of conditions are approximately equal
Checking Sphericity:
Mauchly’s test: p > 0.05 means sphericity is met. - (gives a W statistic)
If sphericity is violated, apply Greenhouse-Geisser (GG) or Huynh-Feldt (HF) which are sphericity corrections.
- We can automatically check the assumption of sphericity when computing the ANOVA test using the anova_test() function from the rstatix package.
77
sphericity corrections
- is a statistic calculated when sphericity assumption is violated
- measures how far the data is from the ideal sphericity
- values range between 0 and 1
- epsilon (E) is the unit we use
- (GGe) for greenhouse geisser epsilon
- (Hfe) for huynh-feldt epsilon
look at greenhouse geisser epsilon first to determine the E value. (GGe)
if E is <.75 we use the greenhouse-geisser(GG) correction
if E is >.75 we use the Huynh-Feldt (HF) correction
78
what would happen without sphericity?
- in danger of making type 2 errors and missing real effects
* If sphericity is violated, the test loses statistical power, which means that the test becomes less sensitive to detecting true differences between conditions
* The test may underestimate differences between conditions, leading to a non-significant result (p > 0.05), even if one rehearsal condition/method is actually better than the others
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Sphericity vs. Homogeneity of Variance:
Sphericity: Concerns differences between conditions. (mauchlys test) - within subjects
Homogeneity: Applies to between-subjects ANOVA for that the three groups being comapred have same variability WITHIN their data points (Levene’s Test).
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Interpreting Results of One-Way Repeated Measures ANOVA
Without assumption violations: Report standard F-value, p value and dfs
With sphericity violation: Use corrected F-values (GG or HF correction), as-well as the generalised effcet size, and dfs
81
effect size for one way repeated measures ANOVA
Generalized effect size (ηG2): Measures overall impact of IV. (all levels together)
Cohen’s d: Measures differences between two specific conditions we are comparing.
82
post hoc tests in one way repested measures ANOVA
= Determines which conditions differ significantly.
- compares two groups at a time to see if their means differ significantly
Bonferroni-adjusted pairwise comparisons:
- reporting adjusted r values
- the t statistic
- the dfs
- cohens d effect size for each of the pairwise comparisons
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difference between post hoc and pairwise comparisons and how to determine how many pairwise comparisons we need
Post hoc test is the name of the general test vs
Pairwise comparison = specific number of comparisons we are making between groups
for both types of one way ANOVA:
the number of pairwise comparisons = k *(k-1) / 2
where k is the number of groups (conditions)
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what is a factorial ANOVA
=A statistical method to examine the effects of two or more independent variables (IVs) with multiple levels or categories on one dependent variable (DV).
- an IV must have an leaste two levels
Example: Time of day (morning, afternoon, evening) and rehearsal strategy (visual, verbal) affecting memory retention.
- helps to identify whether the factors interact or whether their combined effect is different from their individual effects.
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advantaged of a factorial ANOVA
1. Tests the effects of multiple IVs on one DV.
2. More statistically powerful, reducing error variance.
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the two different designs of a factorial ANOVA
1. Complete: All levels of each IV are paired with all levels of every other IV
2. Incomplete: all levels of each IV are not paired with all levels of every other IV
- Conditions are not fully paired due to limitations (e.g., participant availability, not possible to make all comparisons).
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notations meaning in a factorial ANOVA
the notation tells us the number of levels of each single independent variable
2x2: Two IVs, each with 2 levels (e.g., morning/afternoon and visual/verbal).
3x4: Three levels of IV1 and four levels of IV2.
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conditions in a factorial ANOVA
- Conditions are the different combinations of the levels of the IVs in a factorial design.
Each "condition" represents a unique combination of the different levels of the independent variables that participants may be assigned to.
eg
2x2 design has 4 conditions
3 x 4 design has 12 conditons
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the two main features of a factorial design
1. Main effect
= The effect of a single IV on the DV, regardless of the other IV.
- Example: The effect of communication mode (IV1) on user satisfaction (DV), regardless of profile depth (IV2).
2. Interaction effects
= How the combination of two IVs affects the DV, simulating real-world scenarios.
- Example: The effect of communication mode depends on the profile depth.
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Assumptions of a two-way between subject ANOVA
| factorial
1. Levels of Measurement: DV should be interval or ratio data (continuous).
2. Normality: Residuals (the differences between observed and predicted values) should be normally distributed.
- shapiro-wilk test (p>0.05, assmption met)
3. Homogeneity of Variance: Variances should be similar across groups. Checked with Levene’s test. p>0.05 assumption met
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visualisation of two way between subjects anova
| factorial
Boxplot: Visualize the effect of the IVs (e.g., communication mode) on the DV (e.g., user satisfaction), with different levels of the second IV (e.g., profile depth).
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Interpreting Results of two way between subject factorial AVOVA
for main effect:
- Use partial eta squared (ηp²) to interpret effect size
- found under ges column on r output
- report f value, dfs, p value and np2
for interaction effects:
- Examine how the interaction between two IVs affects the DV
- report f value, dfs, p value and np2
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post hoc tests for factorial anova
Bonferroni test: Used to compare differences between the levels of an IV when there are multiple comparisons.
- do this for each variable
