Statistiek 3 voor Pedagogen – Week 1 College

Definition Statistics

• "Statistics is the science of collecting, organizing and interpreting numerical facts, which we call data."

  • Source: Statistiek in de Praktijk, David S. Moore / George P. McCabe, 1994

• Important matters for the application of statistics (“Applied Statistics”):

  • Selecting a sample from a population

  • Deciding whether a sample is representative

  • Descriptive or inferential statistics

  • Measurement levels (NOIR) and types of variables (categorical/quantitative)

  • Selecting the correct statistical analysis

  • Experimental versus non-experimental research design

Methods (Design) & Statistics (Toolkit)

• Important for the application of statistics ("Applied Statistics"):

  • Selecting the correct statistical analysis

Programma Statistiek 3

• 6 hoorcolleges

  • maandag: theorie

• 6 interactieve colleges

  • woensdag: voorbereiding tentamen

• 5 werkgroepen (verplicht)

  • maandag of woensdag: werken aan opdrachten

• Week 7: Q&A sessie (op woensdag)

• Literatuur:

  • Warner (2020) - Applied Statistics II – International Student Edition 3rd Edition (tentamen)

  • Warner (2013) - Applied Statistics – From bivariate through multivariate techniques: - (tentamen) en

  • Agresti & Finlay (2012/2018) – Statistical Methods for the Social Sciences (tentamen)

Hoorcolleges

• Maandag (theorie)

  • Op de campus

  • Introduceren nieuw(e) hoofdstuk(ken)

  • Insluiten nieuwe theorie in de praktijk en in het weekschema

• Woensdag (voorbereiding tentamen)

  • Op de campus

  • Interactief en met korte quizjes

  • Verheldering, voorbeelden en herhaling

Werkgroepen

• Week 2 t/m 6

  • Op de campus

  • Aanwezigheid verplicht

  • Oefenen met SPSS, vragen stellen

  • Inhoudelijke vragen à tutor à discussieforum à Q&A

Toetsing

• Tentamen: 30 meerkeuzevragen:

  • Woensdag 28 Mei 2025

  • 10 vragen over Statistiek 1 & 2 (A&F 2009/2012/2018)

  • 20 vragen over de statistische analyses uit Statistiek 3 (Warner, 2013/2020)

• Eindcijfer = Tentamencijfer

Materiaal en leerdoelen (1)

• Hoorcolleges:

  • Theorie, samenhang, herhaling en samenhang: week 1 t/m 6

  • Q&A in week 7

• Werkgroepen:

  • Maandag en woensdag: oefenen en mogelijkheid vragen te stellen aan tutor

  • Aanwezigheid verplicht, maximaal één afwezigheid toegestaan

• Boek :

  • Theorie: Agresti (2018) Ch. 9 + 12, Warner (2020-II): Ch. 5, 7, 8, 9, 11, 14 of Warner (2013): Ch. 6, 9, 12, 13, 15, 16, 17, 19, 22

  • Practice: comprehension questions at the end of every chapter

• Herhaling Statistiek 1 and 2:

  • Hoorcollege week 1.

  • StatTalk: “Knowledge-clips” (4-5 min) divided per topic.

  • Grasple

• Canvastoetsen/quizzes:

  • Wekelijkse formatieve toets; uit iedere wekelijkse toets wordt één (bewerkte) vraag in het tentamen gebruikt.

  • Oefententamens Statistiek 1 & 2

Materiaal en leerdoelen (2)

• Doelstellingen Statistiek 3:

  • Herhaling statistiek 1+2 (met name de methoden en assumpties), plus nieuwe toevoegingen en het toepassen van deze methoden in de praktijk (SPSS).

  • Ontdek de samenhang tussen de verschillende methoden in het raamwerk van het Generalized Linear Model (GLM), en daarmee…

  • vormt Statistiek 3 een goede basis voor de B-these.

Statistiek in de praktijk: Pedagogische Wetenschappen

• Belang van goed empirisch onderzoek (en daarvoor is statistiek noodzakelijk):

  • “Regression to the mean. It is a statistical fact of life that extreme scores tend to become less extreme upon re-testing, a phenomenon known as regression toward the mean (Kruger, Savitsky, & Gilovich, 1999). Regression to the mean can fool therapists and patients alike into believing that a useless treatment is effective (Gilovich, 1991)."

Overview Lectures: by week

• Recap(itulation) lecture

• ANOVA / Regression

• Factorial ANOVA

• ANCOVA

• Mediation / Moderation

• MANOVA / Repeated Measures

Overview Lectures: by topic

• Recap lecture

• ANOVA

• Fact. ANOVA

• ANCOVA

• MANOVA

• Rep. Measures

• Q&A lecture

• F-test

• Moderation

• TSS

• MSE

• Tukey Contrasts

• Bonferroni

• Sphericity

• DF

• Mediation

• Type II error

• Type I error

• Dummy

Lecture Week 1: Recapitulation

• Overview of het most important concepts in statistics:

  • Descriptive vs inferential statistics

  • Data, population and sample

  • Reliability and validity

  • Variables, measurement levels and range

  • Central tendency-, dispersion-, and position measures

  • Population distribution, sample distribution and sampling distribution

  • Central Limit Theorem and hypothesis testing

• Focus on empirical analyses:

  • Comparison of 2 groups on one quantitative outcome variable (t-test)

  • Comparison of 2 or more groups on one quantitative outcome variable (ANOVA)

  • Determine relation between 2 quantitative variables (regression analysis)

Definition Statistics

• "Statistics is the science of collecting, organizing and interpreting numerical facts, which we call data."

  • Source: Statistiek in de Praktijk, David S. Moore / George P. McCabe, 1994

• A&F: Statistics consists of a body of methods for obtaining and analyzing data, to:

  • Design [research studies that]

  • Describe [the data to]

  • Make inferences based on these data.

  • Descriptive Statistics:

    • Descriptive statistics summarize sample or population data with numbers, tables, and graphs

  • Inferential Statistics:

    • Inferential statistics make predictions about population parameters, based on a (random) sample of data.

Data, population, sample, reliability, validity

• Doing research by means of data: observation of characteristics

  • Population: the total set of participants, relevant for the research question

    • E.g. Population parameter: average hour of self study per week of all students.

  • Sample: a subset of the population about who the data is collected

    • E.g. Sample statistic: average hour of self study per week of a randomly selected sample of 800 students

• Good data is necessary to answer the research question:

  • Reliability (Precision)

  • Validity (Bias)

Variables, measurement levels and range

• Variable: measures characteristics that can differ between subjects

  • Types: behavior-, stimulus-, subject-, physiological variables

  • Measurement scales (NOIR):

    • Categorical/qualitative

      • Nominal unordered categories (eye color, biological sex)

      • Ordinal ordered categories (disagree/neutral/agree)

    • Quantitative/numerical

      • Interval: equal distance between consecutive values (°C)

      • Ratio: equal distance and true zero point (K)

  • Range:

    • Discrete: measurement unit that is indivisible (# brothers/sisters)

    • Continuous: infinitely divisible measurement unit (body height)

Summarized Scale

• Absolute zero point means that the theoretically lowest possible value indicates an absence (value 0)

Descriptive statistics

• In descriptive statistics, 3 dimensions are of importance:

  • Central tendency - “typical observation”

    • Central tendency measures: mean, mode, median …

  • Dispersion - “variability in observations”

    • Dispersion measures: standard deviation, variance, interquartile range

  • Position - “relative position of the observation(s)”

    • Gives information about relative positions of observations: percentile, quartile, …

Sample problems with inferential statistics

• Goal: reliable and valid statements about the population based on a sample:

  • Sample statistic should not differ from population parameter

• Problems:

  • Sampling error - “natural (random) sampling variation”

  • Sampling bias - “selective sampling”

  • Response bias - “incorrect answer”

  • Non-Response bias - “selective participation”

• Important difference between problems concerning reliability (error) and validity (bias).

• Solution:

  • “A random (or other probability) sampling approach of sufficient size that generates data for everyone approached, with correct responses on all items for all subjects.”

Dimensions of distributions

• Population distribution

  • Proportion of students indicating the need for extra support in mathematics.

• Sample data distribution

  • Proportion of students in the sample (here n = 1000) indicating the need for extra support in mathematics.

• Sampling distribution

  • The probability distribution for the sample statistic (proportion/mean/regression coefficient). To interpret as the result of repetitive taking of a sample of size n (here n=1000).

  • Standard deviation of: = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.38 (1-0.38)}{1000}} = 0.015

  • Standard error (σM) estimated by SEM

Central Limit Theorem for sampling distribution

• Empirical rule for normal distribution

  • 68% within ± 1 of the mean

  • 95% within ± 2 of the mean

  • almost 100% within ± 3 of the mean

• Jaccard and Becker (2002):

  • Given a population [of individual X scores] with a mean of μ and a standard deviation of σ, the sampling distribution of the mean [M] has a mean of μ and a standard deviation [generally called the “[population] standard error,” σM] of \frac{σ}{\sqrt{N}} and approaches a normal distribution as the sample size on which it is based, N, approaches infinity. (p. 189)

Types of probability distributions - I

• (Standard) normal distribution à z-statistic

  • Sampling distribution for proportion(s) when H0 holds.

  • (Sampling distribution for mean when H0 holds and when the population standard deviation is known)

• Student’s T distribution(s) à t-statistic

  • Sampling distribution for mean when H0 holds and when the population standard deviation is unknown.

  • Sampling distribution for regression coefficient(s) when H0 holds.

Types of probability distributions - II

• Chi square distribution(s) à χ2-statistic

  • Sampling distribution for squared deviations (in frequencies) of categorical variables when H0 holds.

• Fisher’s distribution(s) à F-statistic

  • Sampling distribution for ANOVA omnibus test of means when H0 holds.

Sampling distribution and hypothesis testing

• Significance-test or hypothesis-test:

  • Method through which you determine, based on the sample, how strong the evidence against a certain hypothesis is and subsequently decide to (not) reject this hypothesis.

• 5 steps of a hypothesis test:

  • Defining assumptions

  • Set up hypothesis

  • Calculate test-statistic (e.g. t-value)

  • Determine p-value

  • Draw conclusion

Type 1 & Type II error

• Probability of a Type I-error (false positive) is determined by:

  • The chosen significance level (α).

• Probability of a Type 2-error (false negative) is determined by:

  • Effect size

  • Sample size

  • Variance (dispersion) in the sample

• The smaller the chosen Type I-error, the larger the acquired Type 2-error, given a certain sample..

Hypothesis-Testing Examples

• Comparison of 2 groups with one quantitative outcome variable (t-test)

• Comparison of 2 or more groups with one quantitative outcome variable (ANOVA)

• Determine the relation between 2 quantitative variables (regression analysis)

Comparison of 2 groups: t-test

• Comparisons between 2 samples:

  • Dependent samples

    • Husbands and wives (e.g. time spent on household activities)

    • Repeated measures: the same person on two different points in time (e.g. extent of depression symptoms before and after therapy)

  • Independent samples:

    • Men and women in randomly selected samples

    • Democrats and Republicans

• Null hypothesis : H0: m1 = m2

• Assumptions of an independent samples t-test:

  • Dependent variable is quantitative and normally distributed (interval/ratio-level)

  • Equal variances for both groups: s21 = s22

  • Independent observations (within and between groups)

Comparison of 2 or more groups: ANOVA

• ANOVA: ANalysis Of VAriance

  • One-way between subjects ANOVA

    • Each participant falls into only one group (e.g. 4 types of stress situations)

    • For each participant there is one observation (e.g. self-reported anxiety)

  • Groups are determined by the levels (categories) of the factor:

    • In this case the number of different stress situations

• Null hypothesis : H0: m1 = m2 = … = mk

• Assumptions for an ANOVA omnibus test:

  • Dependent variable is quantitative and normally distributed (interval/ratio level)

  • Equal variances for all K groups: s12 = s22= … = sk 2

  • ‘Independent observations’ (within and between groups)

ANOVA test-statistic: F-ratio

• ANOVA:

  • F = MSbg/MSwg MS= mean square, bg = between groups, wg = within groups

  • Numerator (MSbg) information about variance in means between groups (M1, M2, … Mk)

  • Denominator (MSwg) information about variance in means within groups

• The F-test is an omnibus test (‘global test’): is there a difference between one or more of the means?

  • An F-test does not show which groups differ!

• F-test signficant? Two ways to test for differences between specific groups:

  • Post hoc (after the fact, after data collection, explorative) à Tukey’s test

  • A priori (planned beforehand, confirmative) à contrasts, regression analysis

Variance analysis: ANOVA Sums of Squares

• Group-indicator = i (i = 1, …, k)

• Participant-indicator = j (j = 1, …, l )

• First partition each deviation (Y{ij} – MY) = total variance

• (Y{ij} – Mi) Unexplained variance within group and (Mi – MY) Explained variance between group components:

  • (Y{ij} – MY) = (Y{ij} – Mi) + (Mi – MY)

• Square each component: (Y{ij} – MY)^2 = (Y{ij} – Mi)^2 + (Mi – MY)^2

• Then sum the squared components across all scores in entire dataset \sum(Y{ij} – MY)^2 = \sum(Y{ij} – Mi)^2 + \sum(Mi – MY)^2

• SS{total} = SS{wg} + SS_{bg}

One-way ANOVA Table

• k = Number of groups

• N = Total number of observations

• df = Degrees of Freedom

Relation between variables: to bivariate statistics

• The univariate (“one variable”) statistics:

  • Measures of central tendency

  • Measures of dispersion

  • Confidence interval mean/proportion

  • Significance test mean/proportion

  • Significance test difference between groups

• Bivariate (“two variables”) statistics is about investigating a possible association between two different variables:

  • Predictor variable or independent variable

  • Outcome variable or dependent variable

OLS- Bivariate Regression

• Other methods used in Statistics 3 ([M]AN[C]OVA) can be related to OLS- regression (together GLM)

• Association between 2 variables

  • E.g.: exam grade (Y) and hours of self study (X)

• Null hypothesis : H0: ρ= 0, H0: b = 0; H0: R = 0

• Assumptions bivariate regression (simple linear regression)

  • Dependent variable (Y) is quantitative and independent variable (X) is quantitative or dichotomous.

  • There is a linear relationship between Y and X.

  • Independent observations.

  • Equal variance of errors.

  • Errors are normally distributed with a mean of 0 for all values of X.

Regression analysis: components

• Assumed functional form for the population:

  • Yi = β0 + βXi + εi

• Regression function: Yi' = b0 + bX_i

  • Yi' predicted value for Yi

  • X_i observed X for person i

  • b_0 intercept

  • b slope

  • Yi– MY total deviation from the mean

  • Yi'– MY predicted part by X_i

  • Yi – Yi' error/residual of the prediction

• SS{total} = SS{residual} + SS_{regression}

• R^2 = \frac{SS{reg}}{SS{total}}

• SE{est} = \sqrt{\frac{\sum (Y - Y')^2}{N-2}} = \sqrt{\frac{SS{residual}}{N-2}} = \sqrt{(1-R^2) * \frac{SS_{total}}{N-2}}

1 example tested in 3 different ways

• Difference between 2 groups: Independent samples t-test

• Difference between 2 groups: OLS bivariate regression

• Difference between 2 groups: one-factor ANOVA (between-subjects)

Verder in week 1:

• Zelfstudie:

  • Agresti (2018) Ch.9 of Warner (2013) Ch.9

  • Canvas quiz week 1