Week 3 - Parametric test assumptions

Question 1

What are the features of a parametric test?

Answer 1

• assess group means
• data must have normal distribution (+CLT)
• unequal variances allowed
• more powerful

Question 2

What are the features of a non-parametric test?

Answer 2

e.g. correlation tests

• assess group MEDIANS
• data doesn’t need to be normally distributed
• can handle small sample size

Question 3

Questions to ask yourself when deciding to use a parametric test or not

Answer 3

• sample size

- best way to measure central distribution (e.g. median or mean?)

Question 4

What are the parametric test assumptions? (4)

Answer 4

1. Additivity and linearity
2. Normality (Gaussian distribution/Bell curve)
3. Homogeneity of variances
4. Independence of observations

Question 5

Describe the assumption of Additivity and linearity

Answer 5

Involves a standard linear model/ equation (describing a straight line)

Question 6

What is the Standard linear model equation

Answer 6

Yi - b0 + b1X1+Ei

Yi= the ith person’s score on the outcome variable

B0= Y-intercept. value of Y when X = 0. point at which the regression line crosses the y-axis

B1 = regression coefficient for the first predictor (B2 for the second predictor).

• Gradient (slope/ rise over run) of the regression
• Direction/ strength of relationship

Ei= the difference between the actual and predicted value of Y for the ith person
- residual/ error

Question 7

What does it mean for data to be linear and additive?

Answer 7

• X1 and X2 predict Y.
• The outcome is a linear function of the predictors (X1 + X2)
• predictors are added together & do not depend on values of other variables in as in a multiplicative model

The outcome Y is an additive combination of the effects of X1 and X2. e.g. as both X1 and X 2 increase, Y increases also

Question 8

True or false:

The outcome Y is an additive combination of the effects of X1 and X2. e.g. as both X1 and X 2 increase, Y increases also

Answer 8

true

Question 9

How can we assess linearity?

Answer 9

• plot observed vs predicted values (symmetrically distributed around diagonal line)
• plot residuals vs predicted values (symmetrically distributed around diagonal line)

Question 10

How to fix non-linear equations?

Answer 10

• nonlinear transformation to variables
• another regressor that is nonlinear - function - polynomial curve
• examine moderators

Question 11

Describe the assumption of Normality

Answer 11

relevant to:

• parameters (sampling distribution)
• residuals/ error terms
• -> confidence intervals around parameter
• -> Null hypothesis significance testing

Question 12

What is Central Limit Theorem (CLT)?

Answer 12

As the sample size increases toward infinity (gets larger), the sampling distribution approaches normal.

–> sample means will be normally distributed thus you don’t need to worry too much about the distribution that the samples came from.

–> distribution of means from many samples and re-samples

–>sample size must be AT LEAST 30

Question 13

For CLT to apply, what size must the sample size be?

Answer 13

At least 30

Question 14

True or false
According to CLT -
Even if the data is not normal, the sampling distribution of the data will be normal

Answer 14

True

Question 15

True or false

Positively skewed data gathers on the left side and scores bunch at the low values with tails pointing to high values

Answer 15

true

Question 16

True or false

Negatively skewed data gathers on the left side and scores bunch at the low values

Answer 16

false - it gathers on the left (e.g. as you grow conditions get “worse” in life)

they bunch at the high values with tails pointing to low values

Question 17

What is kurtosis?

Answer 17

The amount which data clusters in either the tails (ends) or the peak (tallest part) of the distribution

• heaviness of tails

Question 18

Draw the following:
Negative Kurtosis
Positive Kurtosis
Normal distribution

Leptokurtic (heavy tails)
Mesokurtic
Platykurtic (light tails)

Answer 18

draw on paper

Question 19

What are properties of frequency distributions?

Answer 19

• Skewness

- Kurtosis

Question 20

Checking the distribution to determine if the assumption of normality is met is important. Which graphical displays are used to test for normality?

Answer 20

Q-Q plots (dots on straight line = normal)

Histograms

Question 21

What is the name for the software (e.g. JASP) based method for testing for normality?

Answer 21

Shapiro Wilkes Test

Question 22

Describe the Shapiro Wilkes Test and what a p value of <0.05 means

Answer 22

• tests if data is different from normal distribution

- p < 0.05 = data varies significantly from normal distribution thus normality is violated

Question 23

In Shapiro Wilkes Test, what does a p value >0.05 mean?

Answer 23

Data des not vary significantly from a normal distribution thus the normality assumption is not violated

Question 24

Describe the assumption of homogeneity of variance

Answer 24

Assumes all groups or data points have the same or equal variances = the assumption of equal variances

Question 25

What does homoscedasticity mean?

Answer 25

All groups have equal/ similar variances

Question 26

What does hetroscedasticity mean?

Answer 26

All data points/ groups do NOT have equal variances. = unequal variances

Question 27

Define the "error"

Answer 27

The variance from the residual line Error from what we predicted the y would be based on its X value and what we actually observed from the true data

Question 28

Describe the assumption of independence of observation

Answer 28

Assumes that you do not have repeated measures of data. - residuals (errors) are unrelated - assume based on study design

Question 29

According to the assumption of independence of observations, what happens when observations are non-independent?

Answer 29

results in downwardly biased standard errors. (too small) thus incorrect statistica inferences (p values < 0.05 when they should be > 0.05) --> false significant p values ---> this is why it is important to know study design ---> important for mean values of the outcome to come from a different person or other unit (e.g. family, school)

Question 30

What is is an univariate outlier?

Answer 30

outlier when considering only the distribution of the variable it belongs to

Question 31

What is a bivariate outlier?

Answer 31

outlier when considering the joint distribution of two variables - breaking away from the pattern of the association between two variables

Question 32

What is a multivariate outlier?

Answer 32

outliers when simultaneously considering multiple variables.

Question 33

What type of outlier is difficult to asses using numbers or graphs?

Answer 33

multivariate outliers

Question 34

What types of outliers bias the mean and inflate the standard deviation?

Answer 34

Univariate outliers

Question 35

What types of outliers bias the RELATIONSHIP between two variables e.g. change the strength

Answer 35

bivariate outliers

Question 36

What are the three ways to deal with outliers?

Answer 36

REMOVE the case or trim the data TRANSFORM the data CHANGE the score (winsorizing) pulling the data in e.g. biological data (must be transparent about it when reporting results)

Question 37

What are some reasons for transforming data?

Answer 37

1. ease of interpretation - standardisation e.g. z -scores allow for simpler comparisons 2, reducing skewness - closer to normality 3. equalising spread/ improving homogeneity of variances 4. linearising relationships between variables - to fit non-linear relationships into linear models 5. making relationships additive therefore fulfilling assumptions for certain tets

Question 38

Do linear transformations change the shape of the distribution ? What do they change?

Answer 38

No Changes the value of the mean/ SD but shape remains unchanged

Question 39

How do linear transformations work?

Answer 39

- adding constant to each number, x + 1 - converting raw scores to z-scores (x-m)/SD - mean centring, x- m

Question 40

What type of transformation changes the shape of the distribution?

Answer 40

non-linear transformations - Log, log(X) or ln(x) - Square root of x - Reciprocal, 1/x

Question 41

Can you use a log transformation [log(x)] on data with positive values and if you want to reduce positive skew and stabilise variance?

Answer 41

yes

Question 42

When would you use a square root transformation?

Answer 42

- reduce positive skew - stabilise variance - defined for zero/ positive values

Question 43

When would you use a reciprocal transformation?( 1/x)

Answer 43

- reduce impact of large scores - stabilise variance - it reverses the scores so this can be avoided by reversing the scores before transforming 1/ (Xhighest - X lowest)

Question 44

What are the negatives of transforming data?

Answer 44

- non-linear transformations (used to normalise distribution e.g. log, square root, reciprocal ) CHANGE the data & results --> 1 unit increase on the natural log scale might be different - Transformation can hider if wrong transformation applied - Makes interpretation difficult (dealing with raw sores and transformed)