Statistics 2 Notes PDF

Lecture 1- recap What are statistics? - A set of methods and rules for organising, summarising and interpreting information.​ - Statistics can help condense large quantities of information into a few simple figures or statements Researchers are usually interested in a particular grou...

Lecture 1- recap What are statistics? - A set of methods and rules for organising, summarising and interpreting information.​ - Statistics can help condense large quantities of information into a few simple figures or statements Researchers are usually interested in a particular group of individuals or events​ - e.g. children, Americans, left-handers​ **Population:​** - The ENTIRE group of interest (every single individual or event)​ - A [characteristic of a population], e.g. the average age, is called a [parameter]​ - Usually impossible to test a whole population...​ **Sample:​** - A set of individuals selected from the population, usually intended to be representative of the population​ - Make inferences about population based on sample​ - A characteristic (usually a value) of a sample is called a [statistic​] **Descriptive statistics​** - statistical procedures used to simplify and summarise data​ - Graphs, charts, averages etc.​ **Inferential statistics​** - techniques which allow us to study samples and then make generalisations about the population from which they were selected​ - Wilcoxon, chi-square, correlations etc.​ - Problem: is the sample representative of the population? Sampling error.​ **Methods of research** 1. Correlational Research​ - Observational​ - Look for relationship between variables as they exist naturally​ - e.g. personality and executives​ - BUT cannot determine cause and effect​ 2. Experimental Research​ - Can establish cause-and-effect between 2 variables - Researcher manipulates 1 variable to see effect on the other​ - Some control over the research situation to be sure that extraneous variables don't affect - Random assignment​ [Independent Variable:​] - The variable that is manipulated​ [Dependent Variable:​] - The variable that is observed for changes in order to assess the effect of the treatment​ - It depends on the independent variable​ Control:​ - Condition of independent variable that does not receive the experimental treatment​ - No treatment​ - Placebo (neutral) condition​ - Acts as a baseline for comparison​ Experimental Group:​ - Receive treatment​ Confounding variables​ - Uncontrolled variable that is unintentionally allowed to vary systematically with the independent variable​, e.g. different teachers​ - Can sometimes use certain statistical procedures to take this into account​ 3. **Quasi-Experimental Research​** Almost a true experiment"​ 1. Differential research​ - Compare pre-existing groups - e.g. male/female, left/right-handers​ 2. Time-series research​ - Comparison of observations made at one time v. those made at another​ - e.g. "before and after" **Scales of Measurement​** Nominal Scales​ - Data fall into different categories​ - e.g. eye colour, sex, job, political-party affiliation​ Ordinal Scales​ - Sets of categories are organised in an ordered sequence​ - e.g. ranks, ratings on scales (?)​ Interval Scales​ - Ordered categories where all of the categories are intervals of exactly the same size​ - e.g. Fahrenheit temperature scale​ Ratio Scales​ - Interval scale with absolute zero point​ - e.g. length, volume, time​ **[Recap questions that we should know how to answer:]** 1\. Name parametric tests that you remember! 2\. Name non-parametric tests that you remember!​ 3\. What is the difference between parametric and non parametric tests?​ 4\. Which test should you favour: parametric or non parametric tests? Why is this the case?​ 5\. What do correlations generally do?​ 6\. When should you use a Pearson correlation​? 7\. When should you use a Spearmann Correlation?​ 8\. When should you use Kendall's Tau?​ 9\. Do correlations tell us anything about causation? 10\. Write down a research question that could be answered using a correlational design 11\. Which kind of t-tests are there?​ 12\. When do we use which kind of t-test?​ 13\. What do t tests generally do?​ 14\. What are important assumptions that need to be met when using t tests?​  Lecture 2- from slide 14 What's an experiment?​ **Experimental Designs​** - We set up a number of conditions​ - Each condition has different level of variable of interest​ - Compare an outcome variable across these different conditions​ - Did outcome variable differ in each condition?​ **Important design considerations​** - The only thing that differs between conditions​ - Need to rule out that anything else could have caused change in DV​ - Control​ - Randomisation to conditions​ **Design variations​** - How many levels of the IV?​ - Test more than one IV in same experiment​ - Test more than one DV​ - Between-subjects vs. within-subjects​ - Within-subjects: order effects could confound​ Pre- and post-test​ - What if people in our conditions differ before the experiment began?​ - Measure DV before the experiment​ - Then again after giving your manipulation​ **Experimental Manipulations​** Creating the different levels of your IV(s)​ - Situation (e.g. Asch conformity studies, Zeigarnik's interruptions)​ - Task (e.g. test scores taken on computer or paper-and-pencil)​ - Stimuli (e.g. psychophysics work)​ - Instructions (e.g. use mnemonic A, mnemonic B or no mnemonic)​ "Manipulation" -- loaded term​ Manipulation check​ Keep everything else same except that one IV!​ - To make a fair test​ - Avoid confounding variables​ - Rule out that anything except your IV could have caused the change in DV​ **Where to conduct your experiment​** - Lab: control extraneous variables​, all your equipment to hand​, experiment may be somewhat artificial​ - Field: greater ecological validity​, less control over extraneous variables​, create own lab in field setting?​ **Simple example experiment: Two variations​** - Ivan Pavlov: "*Learn the ABC of science before you try to ascend to its summit*"​ - **Hypothesis**: Use of my new mnemonic will improve memory performance​ - **IV**: Participants use new mnemonic or not​ - **DV**: Number of nonsense syllables recalled correctly from list​ - **Stimuli**: List of nonsense syllables presented by memory drum​ **Variation A**: Between-subjects​ Randomise participants to one of two groups​ - Mnemonic group​ - No mnemonic group​ Testing to be under identical circumstances​ - Only use of the mnemonic or not can vary​ Avoid confounding variables!​ - Don't test one group in AM and other in PM!​ Should experimenter be unaware of which group?​ Analysis: Independent-samples t-test​ **Variation B:** Within-subjects​ All participants will do both conditions​ Randomise to which condition they do first​ - Counterbalancing​ - Practice effect could be issue​ May need fewer participants​ Analysis: Related-samples t-test​ **What if I can't manipulate my IV? Quasi-experiments​** - Quasi = "as if", or "almost"​ - Usually between-subjects design​ - Can't randomise to groups​ - E.g. gender, age group, sexual orientation, has diagnosis of schizophrenia or not, etc. ​ - Otherwise same as true experiments EXCEPT​ - Cannot infer causation​ **A development: RCTs​** - Randomised controlled trials​ - Test intervention​ - E.g. effectiveness of new drug​ - Or effectiveness of psychotherapy​ - Gold standard​ - But still have methodological issues​ - High-quality randomisation​ - Strong controls​ - Usually field setting (e.g. hospital, clinic)​ **Analysing data from experiments​** - Usually with a variation of Analysis of Variance (ANOVA)​ - E.g. two group -- use simplified ANOVA (t-test)​ - Exact test depends on within-subjects, between-subjects, or mixed​ - If well-controlled experiment...​ - If differences are statistically significant (*p* \<.05)....​ - We are 95% sure that our IV caused the effect!​ **Recap** Problem with Z Scores​ Degrees of freedom t-tests (independent, paired, one sample) Reporting results (equations) \- just look at the notes from last year it\'s exactly the same **Lecture 3: Correlation and Regression Analyses** - [Experimental vs. Correlational] - Experimental or quasi-experimental designs​ - Test differences between conditions in DV​ (dependent variable) - Correlational designs​ - Test strength of relationship between variables​ - Can be cross-sectional or longitudinal​ - [Design vs. Data collection] - Don't mix up design of your research with your data collection method​ - Data collection methods (not designs!)​: - Questionnaire​ - Lab test on computer using PEBL, Superlab or E-prime​ - Psychometric tests (often on questionnaire)​ - Survey​ - Observations of behaviour​ - [Describing a correlation] - Correlation coefficient (measure strength and direction of two continuous variables) - Pearson's **r** / Spearman's **rho​** - Number between 0 and 1​ - Indicates strength of relationship​ - May be positive or negative​ - Indicates direction of relationship​ - SPSS gives p-value to test if significantly different from 0​ - E.g. r (189) = -.37, p =.031​ - df = 189, r = -0.37, p = 0.031 - Calculate degrees of freedom (df) = n-2 - n (sample size) is given in the SPSS output - [Positive or negative?] - **Correlation** → relationship between 2 quantities such that when one changes, the other does - [Why use a correlational design?] - Non-manipulative studies can be much more naturalistic - In much research it is not possible to manipulate variables - To establish that there is an association before carrying out an experiment to further examine this - You may wish to understand what the strength of the real-life relationship is between 2 variables - To determine what variables might be potentially the most important influences on other variables - You might wish to predict values of one variable from another, e.g. when making selection choices - Developing explanatory models in real life prior to testing aspects of the model in the laboratory - You might wish to understand the structural features of things such as intelligence - You may wish to study how something changes over time - You may wish to understand the temporal direction of associations -- what changes come before other changes - [Correlation is not causation:] - Does A cause B?​ - **Directionality problem​** → arises when a correlation is observed between 2 variables, but it's unclear which variable influences the other---or if they influence each other mutually. - **Third variable problem​** - Occurs when a confounding variable (C) influences both A and B, creating the illusion of a relationship between them. - [Reliability of measures matters!] - Internal consistency​ - **Cronbach's alpha** as best statistic - Correlation is capped by alpha levels​ → alpha levels \"cap\" the range of correlations considered statistically significant. - Low correlation: no relationship there? Or just poor internal consistency of your measures? - Use measures with high Cronbach's alpha!​ Note to self - Suppose you're testing a correlation (r) with N=30 - At a=0.05 might find that a correlation as small as r=0.30 is significant - At a=0.01 might not be significant because threshold for significance is stricter - In small samples, random variation is more pronounced, and it's harder to distinguish a true correlation from random noise. A larger r is needed to confidently claim significance. - In large samples, random variation averages out, so even small correlations can be detected as statistically significant. - [Variation in data matters!] - Data must show variation​ (the more the better​) - All statistics work on variability across data points​ - No variability = no effects seen ​ - Low variability = limits correlation​ - \% of variance explained by a variable: r2 - E.g. r = 0.50, r2 = 0.25 - r = 0.50 indicates a moderate positive correlation between the two variables - r2 = 0.25 means that 25% of the variance in one variable is explained by the other - E.g. r = 0.30, r2 = 0.09 - r = 0.30 indicates a weaker positive correlation than the first example - r2 = 0.09 means that only 9% of the variance in one variable is explained by the other. This suggests a much weaker explanatory relationship compared to the first example. - r → correlation coefficient - r2 → coefficient of determination: represents the proportion of variance in one variable that is explained by the other variable. - A higher r2 = stronger linear association in terms of explained variability - [Characteristics of a relationship:] - **Form** → linear or nonlinear - **Degree** → how well the data fit the form being considered​ - Linear correlation measures how well the data fit on a straight line​ - Varies from -1 to +1 (perfect correlation)​ - A correlation of 0 indicates no fit at all​ Note for self: degree → the strength and direction of the relationship between two variables as measured by the correlation coefficient(r). Provides insight into how closely the two variables are related and whether the relationship is positive or negative.  - Slide 15 -- mathematical equation: pearson **Regressions** - [Taking correlations further] - **Mediator** (A causes B via C)​ → intervening variable - **Moderator** (A causes B only under certain condition / for some people but not others)​ - Making predictions: Regression​ - One variable predicts an outcome: Linear regression​ - Several variables predict an outcome: Multiple regression​ - Add predictor variables step by step: Hierarchical regression​ - Test underlying structure of many variables: Factor analysis​ - Test feasibility of whole causal model/theory: Structural Equation Modelling​ - [Testing a theory by prediction] - Describe, Understand, Predict, Control​ - Given amount of A, can we predict amount of B?​ - With some amount of error (the residual)​ - Linear regression​ - Usually more than one predictor variable​ - i.e. multiple regression​ - If just one predictor variable, there's little advantage over bivariate correlations​ - [Linear regression:] - Variance explained (r2) - Compare r2 for predictors - Which predict significantly​ - How strong is relation with outcome variable?​ - How much variance does each explain?​ - **Best-fit** means the line that has the smallest possible distance from all the points in the scatterplot​ - This allows us to see a summary of the relationship and an estimate of where new data points are likely to fall​ - If there is no correlation, then the best-fit line would be horizontal​ - Different types of regressions: - Multiple regression (simultaneous regression) - Hierarchical regression - Stepwise - [Equation of a straight line:] - **Y = a + bX​** - Y = y axis score​ - a = the 'intercept' or 'constant' which is the point where the regression line crosses the Y-axis (and value of Y when X=0) - b = the slope of the regression line (regression coefficient)​ - X = x axis score​ - Example: - So if the score on X is 12, the equation would be Y = a + b\*12​ - a is the intercept, i.e. where line of best-fit crosses the Y axis​ - b = slope, so if it goes up by 2 then b​ increases by two for every one a ​ - [Multiple regression:] - ​Similar idea​ - Linear regression finds X → Y - Multiple regression looks at the influence of a number of variables on the (dependent variable) response​: X1, X2, X3 → Y - This allows us to estimate scores on a dependent variable by using a set of predictors rather than just one​ - r2 is squared multiple correlation​ ### ### ### ### ### ### ### ### ### ### ### ### ###  ### ### ### **First Image (Three circles: X1, X2, and Y)** 1. **Each circle represents a variable** (X1, X2, or Y), and the overlaps between the circles represent shared variance. - **Shared Variance**: The part of the variability in Y that can be explained by X1, X2, or both. 2. **Key Explanation**: - **a**: Represents the proportion of Y that overlaps with X1 (shared variance between X1 and Y). For example, if X1 explains 18% of the variance in Y, this area corresponds to that 18%. - **b**: Represents the proportion of Y that overlaps with X2 (shared variance between X2 and Y). 3. **Special Case (No overlap between X1 and X2)**: - If X1 and X2 don't overlap at all, their contributions to Y can simply be **added together**. This means the total variance explained in Y is **a + b**. - In reality, this situation is uncommon because predictors often share some variance. ### **Second Image (Three circles with overlap: X1, X2, and Y)** 1. **When X1 and X2 Overlap**: - In real-world data, predictors (X1 and X2) often correlate with each other, so they share some variance (**b**). - This overlap means you cannot simply add the contributions from X1 and X2 to Y, because **b would be double-counted**. 2. **Multiple Regression**: - **Purpose**: Multiple regression allows you to \"partial out\" the overlap (b) to avoid double-counting. - It calculates how much of Y is uniquely explained by X1 (**a**), how much is uniquely explained by X2 (**c**), and how much is shared (**b**). 3. **Outcome**: - Multiple regression ensures that the variance in Y is correctly partitioned: - **a**: Unique contribution of X1 to Y. - **c**: Unique contribution of X2 to Y. - **b**: Shared contribution of X1 and X2 to Y. ### **Key Concept: R² (Squared Multiple Correlation)** - **R²**: Represents the total proportion of variance in Y that is explained by all predictors combined (X1 and X2). - In the case of overlap (as in the second image), R² = a + b + c. ### **Why Is This Important?** This framework is crucial for understanding how multiple regression analyzes relationships between predictors and the dependent variable while accounting for overlaps. It prevents overestimating the predictors\' influence on Y and provides a more accurate model. - [2 main types of multiple regressions:] - **Direct multiple regression** -- enters all the specified predictors, regardless of if they are significant predictors of Y (also called **enter method**) - **Stepwise multiple regression** -- enters the strongest significant predictor, and adjusts all the other predictors to take account of this. It then selects the strongest significant adjusted predictor and enters that and so on. When there are no other significant effects, it stops (also called forward method) - Notes for self: - Starting Point: The model starts with no predictors. - Step 1: Add the predictor that has the strongest correlation with the dependent variable, as long as its p-value is below a predetermined threshold (usually 0.05). - Step 2: After adding the first predictor, check the remaining variables to see which one has the next strongest correlation, and add it to the model, if it\'s statistically significant. - Step 3: This process continues, adding one predictor at a time, as long as each predictor has a significant contribution (p-value below the chosen threshold). - Forward selection adds predictors to the model, but does not remove any once added. - [Interactions in multiple regression​:] - Interactions - Using the direct method will give individual t values for each predictor (and their p values). These effects are equivalent to main effects in ANOVA - Cross-product variables​ → in a direct multiple regression model with cross-product terms, the model includes both the main effects (the individual predictor variables) and the interaction (cross-product) terms. - Curve components​ - Linear regression of Y on predictor X, however the actual relationship might be a curve. ​ - 1\. Simple linear regression​ - 2\. Direct multiple regression using X as first predictor, and square of X (X2) has second.​ - If there are two curves in the relationship, add X3 etc​ - [Why do we do multiple linear regressions?​] - = Measure two or more variables on the same individual (case, object, etc) to explore the nature of the relationship among these variables. There are two basic types of relationships. 1. Cause-and-effect relationships.​ 2. Functional relationships.​ - **Function**: a mathematical relationship enabling us to predict what values of one variable (**Y**) correspond to given values of another variable (**X**). ​ - **Y**: is referred to as the dependent variable, the response variable or the predicted variable.​ - **X**: is referred to as the independent variable, the explanatory variable or the predictor variable. ​ - [Assumptions: ] - [How to check these assumptions?] - Appropriate graphs - Correlations ​ - Formal goodness of fit tests. [Examples slide 35-39] [Slide 44-45:] - The proposed functional relationship will not fit exactly, i.e. something is either wrong with the data (errors in measurement), or the model is inadequate (errors in specification).​ - The relationship is not truly known until we assign values to the parameters of the model.​ - The possibility of errors into the proposed relationship is acknowledged in the functional symbolism as follows: ​y = f(x) + e - e is a random variable representing the result of both errors in model specification and measurement. As in AOV, the variance of is the background variability with respect to which we will assess the significance of the factors (explanatory variables).​ - [The straight line - a conservative starting point​:] - Regression Analysis: the process of fitting a line to data.​ - Sir Francis Galton (1822-1911) \-- a British anthropologist and meteorologist coined the term "regression". ​ - Regression towards mediocrity in hereditary stature - the tendency of offspring to be smaller than large parents and larger than small parents. Referred to as "regression towards the mean".​ [Slide 46 - graph]   **Lecture 4 Notes** **Recap time!!** **T-test** → compares 2 means - **Independent samples:** compares means from 2 different groups - **Paired samples** (aka repeated measures): Compares means from the same group, usually 2 different conditions **But what if there are more than 2 groups or conditions?** **ANOVA** → compares 2 or more means (variance) (ANOVA = analysis of variance) - **Independent samples** (between-subjects)**:** compares means from 2+ different groups to see if the differences are statistically significant. - **Paired samples** (aka repeated measures): same group, different conditions (2+) **Now looking at t-tests and ANOVAS...** **Experimental Design** - **Between-subjects** - Comparison between 2 groups → measuring differences between groups - **Examples:** Independent-samples t-test, between-subjects ANOVA - **Within-subjects** - Interested in differences within the group because of the different conditions → measuring differences between conditions - **Examples:** Paired-samples t-test, within-subjects ANOVA (repeated measures) **Deeper into ANOVAs...** - The Independent variable is called a factor - Individual treatment levels that conditions that make up the factor are called levels - **Example:** factor = learning → levels A, B, C = 3 different learning conditions - You can alter any number of factors in an experiment, but it's best to keep it simple as possible to make interpretation easier - **Single-factor design**: An experiment with a single independent variable (or factor) being manipulated - **Factorial**: An experiment that involves two or more independent variables (factors) being manipulated simultaneously. (2 IV's) - **Null hypothesis:** no difference between the means - **Experimental hypothesis:** difference between the means - ANOVA represented by F-value **Between-Subjects ANOVA** ( 2+ groups) - **F-value**: is the ratio of variance due to the independent variable (between groups) to variance due to error (within groups)  When analyzing **variance in ANOVA**, we break it into two main components: 1. **Between-Treatments Variability:** Variability due to differences between the groups. Including: a. Treatment effect (variation caused by the IV) b. Individual differences (ie differences in age) c. Experimental error (errors that can't be controlled or explained) 2. **Within-Treatments Variability:** Variability due to differences within each group. Including: d. Individual differences e. b\) Experimental error What do these tell us? → **Between-Treatments Variability** tells us whether the differences between groups are meaningful (**e.g.,** due to the treatment). → **Within-Treatments Variability** serves as the \"noise\" or baseline variability to which the between-treatments differences are compared. And why is that important? → The goal of **ANOVA** is to determine whether the between-treatments variability (due to the treatment effect) is large relative to the within-treatments variability (noise from individual differences and experimental error). If it is, we conclude that the treatment had a significant effect. → If **null** hypothesis is **accepted**, then there is no treatment effect → F = 1 → If **experimental** hypothesis is **accepted**, then there is a treatment effect → F \> 1 **Moving onto...** **Post Hoc Tests** → So, ANOVA tells you **if there's a significant difference somewhere** among the group means by comparing the overall variability between and within groups. → However, ANOVA doesn't tell you ***which*** pairs of group means differ significantly (e.g., whether Group A differs from Group B, or Group C from Group D). → This is where post-hoc tests come in: they help make pairwise comparisons to determine exactly which means are different. → **Several different types:** Tukey's HSD, Fisher's LSD, Bonferroni **Example: Between Subjects Anova + Tukey's** 1. **Descriptive Statistics →** Summarize the group means and variability. - Summarizes key metrics - mean, sample size (N), and standard deviation for each group. - Shows differences between group means (Group A has the highest mean, and Group C the lowest). - These differences suggest there may be variability between groups due to the treatment effect, but further testing (ANOVA) is needed to confirm statistical significance. 2. **ANOVA** → Test for an ***overall*** significant difference among the groups.  - What it shows: - Breaks down the variability in the data into between-groups variability (due to the treatment) and within-groups variability (due to individual differences or experimental error). - Provides the test statistic (F) and p-value to assess whether the group means differ significantly. - Equation: **F(2,27) = 10.24, p\ 0.05), or 1 and 3 (p \> 0.05) **What are the assumptions of an ANOVA?** - **Equal Variances (Homogeneity of Variance):** - The variances of the populations being compared (across groups) should be approximately equal. - This is tested using Levene\'s Test or similar procedures. - Violation of this assumption can lead to biased results. - **Normality:** - The populations from which the samples are drawn should follow a normal distribution. - This is most critical when sample sizes are small. For larger sample sizes, the Central Limit Theorem often mitigates this issue. - **Independence of Observations:** - Each observation must be independent of others within and across groups. - This means no participant's score should influence another's. **Non-Parametric ANOVA Equivalents** When the above assumptions are violated (e.g., data are not normally distributed or variances are unequal), you can use non-parametric tests: - **Kruskal-Wallis H Test** (for Between-Subjects ANOVA)**:** - This test compares the medians of three or more independent groups. - It is based on rank data and does not assume normality or equal variances. - **Friedman Test** (for Within-Subjects/Repeated Measures ANOVA)**:** - This test compares the medians of three or more related groups (e.g., the same participants measured under different conditions). - Like Kruskal-Wallis, it works with ranked data and avoids assumptions of normality or equal variances. **What should I do if I want to know the impact of the combination of multiple Independent variables?** **Factorial ANOVA** → uses more than one categorical independent variable (e.g.two-way ANOVA) → answers how do 2 or more independent variables affect a response variable → Factorial ANOVA helps us understand how multiple factors (and their interactions) affect a dependent variable. **Simplest form of factorial ANOVA:** - Two factors (each being binary, meaning they have two levels) four distinct groups to compare - **Example:** effect of study method (visual vs. auditory) and test timing (morning vs. evening) on test scores → we'd have four groups: visual-morning, visual-evening, auditory-morning, and auditory-evening. - Explores the interaction between factors (whether the impact of one factor depends on the level of the other factor) - E.g.: Maybe the visual method works better in the morning, but the auditory method works better in the evening **THE END** **Good job!! 👏** **Revision Questions (from Lecture 4)** 1. Which Epistemology is used in statistics? **Positivism** → school of research thought that sees observable evidence as the only form of defensible scientific findings. 2. What is a sample? A group of people being measured that represents a population → taken from a bigger population → represents the bigger population 3. What are inferential statistics? → making inferences about data (relates to variables) → always relating something (relating 2 parts of info) 4. What is correlational design? → testing the relationship between 2 variables → observing (not experimenting) 5. What are possible solutions for replication crisis in quantitative research → data being publicly available → opening science 6. Which type of analysis would you use to compare the means of 2 groups? (data = normal distribution and ratio scaled) → Independent t-test 7. What are 3 characteristics of experimental research? 1. Controlled setting 2. Control group and experimental group 3. Cause and effect 8. How can we test for significant differences between 4 groups of data? (data is ratio scaled and normal distribution) → ANOVA 9. Which test would you use to answer the following: 'To what extent can age, gender, education level, left and right-handedness, and schooling predict IQ'? → multiple regression analysis (estimates the linear relationship between a scalar response and one more explanatory 10. When can you not randomize? 11. Which test would you use to answer the following: 'How do items in a questionnaire tap onto the same underlying construct?' 12. What is the null hypothesis? 13. What is the experimental hypothesis? 14. What do Post hoc tests allow us to identify? 15. Compare and contrast t-tests and ANOVAs? **Lecture 5** [What is a factor analysis?] - A family of techniques to examine correlations amongst variables - Uses correlations among many items to search for common clusters - Aim is to identify groups of variables which are relatively homogeneous - Groups of related variables are called '**factors**' - Involves empirical testing of theoretical data structures​ - [Purposes: ] - The main applications of factor analytic techniques are: ​ 1. to reduce the number of variables 2. to detect structure in the relationships between variables, that is to classify variables. → method that examines correlations in search of clusters of highly correlated variables. In order to reduce complexity of data by data reduction. As shown in the diagram. - [History of factor analysis?] - Invented by Spearman (1904) - Usage hindered by the difficulty of hand calculation - Since the advent of computers, usage has thrived, especially to develop: - Theory → e.g. determining the structure of personality - Practice → e.g. development of 10,000s+ of psychological screening and measurement tests - [Example: factor analysis of essential facial features] - 6 factors represent 76.5% of the total variability in facial recognition. They are (in order of importance): upper-lip, eyebrow-position, nose-width, eye-position, eye / eyebrow-length, face-width - [Problems with factor analysis include:​] - Mathematically complicated ​ - Technical vocabulary​ - Results usually absorb a dozen or so pages​ - Students do not ordinarily learn factor analysis ​ - Most people find the results incomprehensible​ - [Exploratory vs. Confirmatory Factor Analysis​:] - **EFA = Exploratory Factor Analysis​** - Explores & summarises underlying correlational structure for a data set​ - **CFA = Confirmatory Factor Analysis​** - Tests the correlational structure of a data set against a hypothesised structure and rates the "goodness of fit"​ - [Data reduction 1] - Factor analysis simplifies data by revealing a smaller number of underlying factors​ - Factor analysis helps to eliminate:​ redundant variables, unclear variables, irrelevant variables​ - [Steps in factor analysis:] 1. Test assumptions​ 2. Select type of analysis​ (extraction & rotation)​ 3. Determine number of factors​ 4. Identify which items belong in each factor​ 5. Drop items as necessary and repeat steps 3 to 4​ 6. Name and define factors​ 7. Examine correlations amongst factors​ 8. Analyse internal reliability​ - [Assumption Testing -- Factorability:] - It is important to check the factorability of the correlation matrix ​ (i.e., how suitable is the data for factor analysis? → whether the data is factorable, meaning that the variables in your dataset can be grouped into factors) - Check correlation matrix for correlations over 0.3​ - Check the anti-image matrix for diagonals over 0.5​ - Check measures of sampling adequacy (MSAs)​ - Bartlett's - KMO​ - The most manual and time consuming but thorough and accurate way to examine the factorability of a correlation matrix is simply to examine each correlation in the correlation matrix​ - Take note whether there are SOME correlations over 0.30 ​ \*\*The **correlation matrix** is a table that shows the correlation coefficients between all pairs of variables in your dataset. For factor analysis to work effectively, there should be sufficient correlation between the variables\*\* - [Extraction Method:​ PC vs. PAF​] - There are two main approaches to EFA based on:​ 1. Analysing only shared variance​ → **Principle Axis Factoring (PAF)​** 2. Analysing all variance​ → **Principal Components (PC)​** - [Principal components (PC)] - More common​ - More practical ​ - Used to reduce data to a set of factor scores for use in other analyses​ - Analyses all the variance in each variable​ - [Principle Axis Factoring (PAF)] - Used to uncover the structure of an underlying set of p original variables​ - More theoretical​ - Analyses only shared variance​ (i.e. leaves out unique variance)​ - [PC vs. PAF] - Often there is little difference in the solutions for the two procedures - Often it's a good idea to check your solution using both techniques​ - If you get a different solution between the two methods​ - Try to work out why and decide on which solution is more appropriate​ - [Communalities: ] - The proportion of variance in each variable which can be explained by the factors​ - Communality for a variable = sum of the squared loadings for the variable on each of the factors​ - Communalities range between 0 and 1 - **High communalities (\>.5)** show that the factors extracted explain most of the variance in the variables being analysed​ - **Low communalities (\<.5)** mean there is considerable variance unexplained by the factors extracted​ - May then need to extract MORE factors to explain the variance​ - **[Eigen values]** - EV = sum of squared correlations for each factor​ - EV = overall strength of relationship between a factor and the variables​ - Successive EVs have lower values​ - Eigen values over 1 are 'stable'​ - [Explained variance] - A good factor solution is one that explains the most variance with the fewest factors - Realistically happy with 50-75% of the variance explained​ - **How many factors?** → determining how many factors is a subjective process → one should seek to explain maximum variance using fewest factors, considering: 1. Theory: what is predicted/expected? 2. Eigen values \> 1? (Kaiser's criterion) 3. Scree Plot - where does it drop off? 4. Interpretability of last factor? 5. Try several different solutions? 6. Factors must be able to be meaningfully interpreted and make theoretical sense → one should: - Aim for 50-75% variance explained with ¼ to ⅓ as many factors as variables/items - Stop extracting factors when they no longer represent useful/meaningful clusters or variables - Keep checking/clarifying the meaning of each factor and its items **Scree Plot** → a bar graph of eigenvalues - It depicts the amount of variance explained by each factor - Look for the point where additional factors fail to add appreciably to the cumulative explained variance - The 1st factor explains the most variance - The last factor explains the least amount of variance **Initial Solution - unrotated factor structure 1** - Factor loadings (FLs) indicate the relative importance of each item to each factor - In the initial solution, each factor tries 'selfishly' to grab maximum unexplained variance - All variables will tend to load strongly on the 1st factor - Factors are made up of linear combinations of the variables **Initial Solution - unrotated factor structure 2** - 1st factor extracted: - Best possible line of best fit through the original variables - Seeks to explain maximum overall variance - A single summary of the main variance in a set of items - Each subsequent factor tries to maximise the amount of unexplained variance which it can explain.​ - Second factor is orthogonal to first factor - seeks to maximize its own eigenvalue (i.e., tries to gobble up as much of the remaining unexplained variance as possible) **Initial Solution - unrotated factor structure 3** - You seldom see a simple unrotated factor structure​ - Many variables will load on 2 or more factors​ - Some variables may not load highly on any factors​ - Until the FLs are rotated, they are difficult to interpret.​ - Rotation of the FL matrix helps to find a more interpretable factor structure. **Two basic types of rotation:** - **Orthogonal rotation:** factors are kept uncorrelated. Orthogonal rotation is simpler and easier to interpret, but it may not capture the true structure of the data if the factors are actually related. ​ - **Oblique rotation**: factors have some degree of correlation. Oblique rotation is more realistic and flexible, but it may introduce complexity and ambiguity in the factor interpretation. **Orthogonal versus Oblique Rotations** (which to use) - Think about purpose of factor analysis​ - Try both​ - Consider interpretability​ - Look at correlations between factors in oblique solution - if \>.32 then go with oblique rotation (\>10% shared variance between factors) - No option for Oblique in PSPP **Two basic types of factor rotation:** - **Orthogonal:** minimises factor covariation, produces factors which are uncorrelated - **Oblimin​:** allows factors to covary, allows correlations between factors. **Orthogonal Rotations in PSPP** → Varimax, Quartimax, Equimax in PSPP​ - **Variamax:** maximizes the variance of the factor loadings within each factor, which means that it tries to make the loadings as high as possible for some variables and as low as possible for others = clear and distinct factors that are easy to label and interpret. ​ - **Quartimax** minimizes the variance of the factor loadings within each factor, which means that it tries to make the loadings more similar for all variables = factors that are more general and broad, but less specific and meaningful​ - **Equamax** rotation is a compromise between varimax and quartimax, which balances the variance of the factor loadings within and across factors.  - **Factor loading** (numbers on the pattern matrix) = how strongly each variable correlates with a given factor - **Components 1,2,3** = extracted factors - Each row is a psychological trait, and the numbers show how strongly that trait loads onto each factor (factor loading) - **Rotated Solution:** The use of Oblimin rotation suggests that the factors are allowed to correlate rather than being independent. - **Takeaways:** - Factor analysis groups related psychological traits into broader categories (latent constructs). - Different rotations can slightly shift how traits are associated with factors. - Cross-loadings (like the circled value in diagram 1) indicate that some traits do not fit neatly into one factor. **Factor Structure** → Factor structure is most interpretable when:​ 1. Each variable loads strongly on only one factor​ 2. Each factor has two or more strong loadings​ 3. Most factor loadings are either high or low with few of intermediate value​ 4. Loadings of +.40 or more are generally OK **How to eliminate items:** → A subjective process, but consider:​ - Size of main loading (min=.4)​ - Size of cross loadings (max=.3?)​ - Meaning of item (face validity)​ - Contribution it makes to the factor​ - Eliminate 1 variable at a time, then re-run, before deciding which/if any items to eliminate next​ - Number of items already in the factor **How many items per factor:** - More items in a factor → greater reliability​ - Minimum = 3​ - Maximum = unlimited​ - The more items, the more​ rounded the measure​ - Law of diminishing returns​ - Typically 4 to 10 is reasonable **Interpretability** - Must be able to understand and interpret a factor if you're going to extract it​ - Guide by theory and common sense in selecting factor structure​ - However, watch out for 'seeing what you want to see' when factor analysis evidence might suggest a different solution​ - There may be more than one good solution, e.g.,​ - 2 factor model of personality​ - 5 factor model of personality​ - 16 factor model of personality​ **Factor Loadings and Item Selection 1** → Factor structure is most interpretable when:​ 1. each variable loads strongly on only one factor (strong is \> +.40)​ 2. each factor shows 3 or more strong loadings, more = greater reliability​ 3. 3\. most loadings are either high or low, few intermediate values​ → These elements give a 'simple' factor structure. **Factor Loadings and Item Selection 2** - Comrey & Lee (1992)​ - loadings \>.70 - excellent​ - \> 63 - very good​ - \>.55 - good​ - \>.45 - fair​ - \>.32 - poor - Choosing a cut-off for acceptable loadings:​ - look for gap in loadings​ - choose cut-off because factors can be interpreted above but not below cut-off **CHATGPT QUESTIONS** **Basic Concepts of Statistics** **Q:** What is the difference between a population and a sample?\ **A:** A population includes every individual of interest, while a sample is a subset selected to represent the population. **Q:** What is the main purpose of descriptive statistics?\ **A:** To organize, summarize, and present data clearly. **Q:** How do inferential statistics differ from descriptive statistics?\ **A:** Inferential statistics make generalizations from a sample to a population, while descriptive statistics summarize data. **Q:** What is a parameter, and how is it different from a statistic?\ **A:** A parameter describes a population, while a statistic describes a sample. **Q:** What is sampling error, and why is it important?\ **A:** The difference between a sample statistic and a population parameter; it affects the accuracy of conclusions. **Research Methods** **Q:** What is the key difference between correlational and experimental research?\ **A:** Correlational research examines relationships, while experimental research establishes cause-and-effect. **Q:** What is the independent variable in an experiment?\ **A:** The variable manipulated by the researcher. **Q:** What is the dependent variable in an experiment?\ **A:** The variable measured to assess the effect of the independent variable. **Q:** What is the purpose of a control group in experimental research?\ **A:** To serve as a baseline for comparison. **Q:** Why is random assignment important in experimental research?\ **A:** It reduces bias and ensures equal distribution of variables. **Statistical Tests** **Q:** What is the difference between parametric and non-parametric tests?\ **A:** Parametric tests assume normality, while non-parametric tests do not require normal distribution. **Q:** When would you use a Pearson correlation instead of a Spearman correlation?\ **A:** Pearson is used for linear relationships with normally distributed data; Spearman is for ordinal data or non-linear relationships. **Q:** What are the three main types of t-tests, and when should each be used?\ **A:** Independent samples t-test (two different groups), paired t-test (same group under different conditions), one-sample t-test (compare sample mean to population mean). **Q:** What is an ANOVA, and why is it used instead of multiple t-tests?\ **A:** ANOVA tests for differences between three or more groups, reducing the risk of Type I errors from multiple t-tests. **Q:** What does the F-value in an ANOVA test tell us?\ **A:** It measures the ratio of variance between groups to variance within groups. **Post Hoc Tests and Assumptions** **Q:** What do post hoc tests allow researchers to determine?\ **A:** Which specific groups differ after finding a significant ANOVA result. **Q:** Name three commonly used post hoc tests in ANOVA.\ **A:** Tukey's HSD, Bonferroni, Fisher's LSD. **Q:** What are the key assumptions of a t-test?\ **A:** Normality, homogeneity of variance, independence of observations. **Q:** What are the three main assumptions of an ANOVA?\ **A:** Normality, homogeneity of variance, independence of observations. **Q:** When would you use a non-parametric alternative to an ANOVA, such as the Kruskal-Wallis test?\ **A:** When data are not normally distributed or variances are unequal. **Factor Analysis** **Q:** What is the goal of factor analysis?\ **A:** To identify underlying factors that explain patterns in data. **Q:** What is the difference between Exploratory Factor Analysis (EFA) and Confirmatory Factor Analysis (CFA)?\ **A:** EFA identifies unknown factor structures, while CFA tests a hypothesized structure. **Q:** What is an eigenvalue, and what does it represent?\ **A:** It represents the variance explained by a factor in factor analysis. **Q:** What does a scree plot help determine?\ **A:** The number of factors to retain in factor analysis. **Q:** What is the difference between orthogonal rotation and oblique rotation in factor analysis?\ **A:** Orthogonal rotation assumes factors are uncorrelated, while oblique allows correlation. **Regression Analysis** **Q:** What is the main purpose of a linear regression?\ **A:** To predict one variable based on another. **Q:** How does multiple regression differ from simple regression?\ **A:** Multiple regression uses multiple predictors, while simple regression uses only one. **Q:** What does R² tell us in regression analysis?\ **A:** The proportion of variance in the dependent variable explained by the predictors. **Q:** What is a mediator variable, and how does it differ from a moderator variable?\ **A:** A mediator explains the relationship between two variables, while a moderator affects the strength of the relationship. **Q:** What are the main assumptions of a regression model?\ **A:** Linearity, no multicollinearity, independence, homoscedasticity, and normality. **Miscellaneous** **Q:** What does the phrase "correlation does not imply causation" mean?\ **A:** Just because two variables are related does not mean one causes the other. **Q:** What is the replication crisis, and how can it be addressed in research?\ **A:** The difficulty of replicating scientific findings; solutions include open data and pre-registration of studies. **Q:** What is the role of degrees of freedom in hypothesis testing?\ **A:** It affects the variability and significance of test results. **Q:** Why might researchers use hierarchical regression instead of standard multiple regression?\ **A:** To test the predictive power of variables in a stepwise manner. **Q:** What is the positivist epistemology in statistics?\ **A:** A philosophy that emphasizes objective, observable evidence in scientific research.