Statistical tests- lecture slides

Question 1

What is a two-way ANOVA test used for?

Answer 1

To test the interaction effects between two independent categorical variables on a continuous dependent variable.

Question 2

What test would we use to compare two related measurements?

Answer 2

A Paired t-test

Question 3

What test would we use to compare three or more independent groups?

Answer 3

One-way ANOVA

Question 4

What test would we use to examine two factors together?

Answer 4

Two-way ANOVA

Question 5

What is the Normal Distribution?

Answer 5

A bell-shaped curve where most values cluster around the mean.
Symmetrical, with equal probability of values occurring above or below the mean.

Question 6

What is Central Limit Theorem?

Answer 6

As sample size increases, sample means approximate a normal distribution, regardless of the population’s shape. This is important as many statistical tests assume normality and it allows us to make inferences about populations from sample data.

Question 7

Non-parametric tests

Answer 7

Wilcoxon, Kruskal-Wallis

Question 8

What do t-tests do?

Answer 8

t-statistic (t)
Compares two groups (e.g., test scores of Group A vs. Group B).
A bigger t-value means a bigger difference between groups

Question 9

What do ANOVA tests do?

Answer 9

F-statistic (F)
Compares three or more groups.
A bigger F-value means more variation between groups than within groups.

Question 10

Common transformations

Answer 10

Log Transformation: Used when data is right-skewed (e.g., income, rainfall).
Square Root Transformation: Useful for count data, left-skewed data.

Question 11

Key points of CLT

Answer 11

Larger samples reduce variability in sample means.
Sample means from a large enough sample will always approximate a normal distribution.
The CLT allows us to use statistical tests even if the population is not normally distributed → fundamental to statistical inference.
Practical Implication: Even with non-normal data, large sample sizes ensure the validity of hypothesis testing.
This principle underpins many statistical tests used in social sciences.

Question 12

What is the Shapiro-Wilk Test

Answer 12

A statistical test that checks if a dataset is normally distributed.
Compares the actual data distribution to a perfect normal distribution.
Interpretation: p > 0.05 → Data is likely normal ✅; p ≤ 0.05 → Data is not normal ❌

Question 13

What is the Levene’s Test

Answer 13

A test that checks if different groups have similar variances (spread of data).
This is important as T-tests & ANOVA assume equal variances between groups.
If variances aren’t equal, results may be misleading.
Interpreting Results:
p > 0.05 → Variances are equal ✅ (Safe to use t-test or ANOVA).
p ≤ 0.05 → Variances are not equal ❌ (Use Welch’s t-test or Welch’s ANOVA)

Question 14

When should you use a paired t-test?

Answer 14

Comparing two related measurements within the same group.

Question 15

Key assumptions of a paired t-test

Answer 15

Data is normally distributed.
Observations are dependent (paired data).
If assumptions fail: Wilcoxon Signed-Rank Test (non-parametric)

Question 16

When should you use a one-way ANOVA

Answer 16

Comparing means across three or more independent groups.

Question 17

Key assumptions of a one-way ANOVA

Answer 17

Normality in each group.
Homogeneity of variances (Levene’s test).
Observations are independent.
Alternatives if Assumptions Fail: Kruskal-Wallis Test, Welch’s ANOVA.

Question 18

Properties of Kruskal-Wallis test

Answer 18

A non-parametric test.
Compares median ranks rather than means.
Used when data is not normal, variances are unequal, and when using ordinal data.

Question 19

Properties of Welch’s ANOVA

Answer 19

A parametric test.
Compares means.
Used when normality assumed but variances are unequal.

Question 20

When should you use a two-way ANOVA?

Answer 20

Evaluating the effect of two independent variables simultaneously

Question 21

Key assumptions of a two-way ANOVA

Answer 21

Normality in each group.
Homogeneity of variances.
No significant interaction between variables unless tested.

Alternatives if Assumptions Fail: Generalised Linear Models (GLM)

Question 22

When is a Chi Square test used? + properties

Answer 22

Used where nominal variables are being compared.
There are two or more categories for each variable.
Non-parametric.
The test measures the extent to which observed data departs from expectation

Question 23

Results in a Chi-square test

Answer 23

Where chi-square is below 0.05 there is a high probability that the variables are correlated.
Results are between 0 and 1 where:
0 shows no association between the two variables
1 shows a perfect association between variables.
The higher the value, the stronger the association.

Question 24

How do you calculate Pearson’s correlation in R?

Answer 24

> cor.test (variable1, variable2)
Pearson’s product-moment correlation data:

Question 25

How do you calculate Spearman's correlation in R?

Answer 25

> cor.test(asthma$Asthma, asthma$PM10, method="spearman") Spearman's rank correlation rho data:

Question 26

What does regression measure?

Answer 26

Association where a causal relationship is believed to exist e.g. based on scientific studies. Compares one ‘dependent’ variable and one or more ‘independent’ variables. E.g. regression rather than correlation should be used when comparing lung cancer rates (dependent variable) vs. number of cigarettes smoked per day (independent).

Question 27

Correlation

Answer 27

Allows you to identify some information about a relationship: the direction and significance

Question 28

Regression

Answer 28

Allows you to model the relationship between the variables in detail: how much does x change with y?

Question 29

When should you use linear regression?

Answer 29

When understanding more than the strength and direction of a relationship between two continuous variables When your data meet the necessary assumptions

Question 30

Assumptions for linear regression models about the nature of the data/residuals

Answer 30

The data are independent The measurement scale of the data is interval or ratio (not categorical) The relationship between the variables is linear (but there are ways around this) There is no significant measurement error in the x variable The variance in the residuals is constant (the model fit is similar across the data) The residuals are normally distributed

Question 31

How do you build a linear model in R?

Answer 31

Dependent ~ Independent (or Response ~ Explanatory) y ~ x y = mx + c c = constant or intercept – your model will include an intercept (unless you specify otherwise) m = slope, temperature lm(formula = Biomass ~ temperature, data = data)

Question 32

What do the stars mean in R?

Answer 32

They correspond to the level of significance (the smaller the decimal the more significant)

Question 33

What do residuals measure?

Answer 33

The distance from your line of best fit to your data

Question 34

The smaller the residuals..

Answer 34

..the closer the line of best fit to the data

Question 35

What must residuals be?

Answer 35

Normally distributed, equal of variance throughout

Question 36

What does no pattern in the residuals mean?

Answer 36

The model is succeeding in capturing the pattern in the data

Question 37

What can you do if it is not appropriate to use a linear model for a dataset?

Answer 37

Log-transforming- transformations express the same data on a different scale. lm(Biomass ~ log(Temperature), data = data)

Question 38

What do logistic models model

Answer 38

The log-odds of an event as a linear combination of one or more independent variables

Question 39

What is it called when there are correlated variables

Answer 39

collinearity

Question 40

What must you test for when there are many explanatory variables

Answer 40

multicollinearity- collinearity between all variables

Question 41

VIF (variation inflation scores)

Answer 41

over 5 is problematic, over 10 should not be included in the model

Question 42

ANCOVA

Answer 42

analysis of COvariance a linear relationship with multiple levels or 'treatments' (categorical data) e.g. lm(SLA ~Altitude + Veg_type, data = data)