ISA Summary

Chapter 1

Empirical questions: questions that pertain to knowing “what is” – based on observation/experiment

Normative questions: questions that pertain to “what ought to be”

Scientists: people who engage in collecting and interpreting empirical information

Hypotheses: statements positing possible relationships or associations among the phenomena being studied

Social sciences: the empirical study of social phenomena (share two common aspects with all sciences – scientific method & the use of quantitative techniques for gathering information)

Scientific method: a series of logical steps that, if followed, help minimise any distortion of facts stemming from the researcher’s personal values and beliefs – involves formulating hypotheses, testing of hypotheses, and verification of hypotheses

(Contingency) table/Cross-tabulation: a way of presenting data for purposes of testing hypotheses

Cell: intersection of a particular row and a particular column

Associated: when a case fall into a particular category of one concept and it also falls into a particular category of another

Concept: ideas

Variables: concepts that vary, or change, from one observation to another

Induction: a process of reasoning that goes from the specific to the general

Deduction: a process of reasoning that goes from the general to the specific

Experiment: a test of a hypothesis under laboratory-like conditions

Scientific laws: hypotheses verified so often that they have a high probability of being correct

Data: all the information we use to verify a hypothesis

Datum/ piece of data: a single piece of information

Necessary condition: a condition that must be present in order for some outcome to occur

Sufficient condition: a condition in which the predicted outcome will definitely take place

Theory: a set of interrelated hypotheses that together explain some phenomenon

Main diagonal: diagonal line in a table from upper left to lower right

Positive relationship: a relationship in which greater is associated with greater; less with less

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Off-diagonal: clustering on a diagonal line in a table that goes from the upper right-hand side to the lower left-hand side

Inversely related: a condition in which most cases cluster about the off diagonal – a high score on one variable is associated with a low score on the other

Statistics: the study of how we describe and make inferences (conclusions) from data

Cause: when one phenomenon being studied brings about the other – needs association and temporal sequence: when one phenomenon being studied occurs earlier in time than the other (cause before effect)

Dependent/criterion variable: the variable that is being caused or explained (e.g. performance)

Independent/ predictor variable: the variable that is doing the causing/explaining (e.g. gender)

Unit of analysis: what we actually measure or study to test our hypothesis: from whom or from what the measurement is made

Chapter 2

Data: information specifically intended to assist decision making or to provide analysis

Measurement: a very specific process (e.g. measuring length) or simpler actions (e.g. assigning people into categories/variables)

Levels of measurement: measurement that falls into one of four general categories:

N ominal: Categorisation into groups/ No ranking O rdinal: Ranking along given dimension-Distance between categories unequal unknown I nterval: Ranking along a given dimension / Distance between categories equal /

No meaningful or absolute zero R atio: All properties of interval / Absolute and meaningful zero point

Qualitative/Categorical data: data that are assignment to categories but do not imply amounts (e.g. major of college students, gender, etc.) – nominal level

Quantitative data: data that are assigned to categories that are involved with amounts (e.g. age, income, etc.) – interval/ratio level

Nominal levels of measurement:

Nominal: pertains to the act of naming Attributes: characteristics that do not necessarily involve amounts (e.g. gender, eye colour,

religion, etc.) – part of nominal level of measurement Individual form of data: data that are presented in a list of each individual subject in the

study and his/her category assignment for one or more variables Grouped nominal data: data that are presented as a category of the variable listed – the

subjects are not named by are counted (grouped) in the category into which each subject falls Frequency distribution: a tabulation that lists the variable, its categories and a frequency

column (e.g. Male f=4 – Female f=6 – n= 10)f= frequencyn= total number of cases (sum of frequency)

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Dichotomies: two-category variables (e.g. gender= nominal dichotomy) N-category: a term indicating more than two categories

Ordinal levels of measurement:

Ordinal: involving a rank order or other ordering Individual ordinal: the ranking of each individual subject from highest to lowest along the

variable Grouped ordinal data: data that present subjects placed into ranked categories, order highest

to lowest (or lowest to highest) Likert scales: a scale whose categories are based on the level of agreement with a particular

statement or issue (e.g. Strongly Disagree – Disagree – Agree – Strongly Agree)

Frequencies: headcounts or tallies indicating the number of cases in a particular category / the total number of cases measured (e.g. 8 cloudy days, 11 rainy days, etc.)

Scores: numbers that are used to represent amounts or rankings ( 1st: John, 2nd: Peter 3rd: Alan – represent tallest, second tallest, third tallest, etc.)

Interval & Ratio levels of measurement:

Interval: an interval scale in which the subject receives a numerical score (rather than ranking) and where zero is a random chosen point, rather than lack of what is being measured (e.g. IQ) – scores may be below zero as well as above

Forms of Interval-level data:- Individual listings/Raw scores: a simple numerical score / original scores that has not

been transformed into percentage or average- Ungrouped frequency distribution: scores listed in a sequence (usually highest to

lowest) that includes every score that appears in the results- Grouped interval data: grouped data that are also at the interval level of measurement

Example: AGE 10-14.9 = 615- 19.9 = 1020- 24.9 = 1025-25.9 = 4 Class interval: an interval that indicates the space between two end points

Two criteria for data to remain at the interval level of measurement:- The class intervals must all be equal in size- All class intervals must be close-ended: each class must have both an upper and lower

limit – otherwise it is at the ordinal level of measurement (Open-ended intervals: a class interval that has a lower limit but no upper limit, or vice versa – e.g. ’25 and above’, ‘below 10’)

Ratio: a level of measurement similar to interval level, but where zero is an absolute zero, meaning none of what is being measured – scores may not be below zero (e.g. age)

Absolute zero: a zero that means a complete lack of the variable being measured rather than some random chosen point

Demographic variables: background information on the human subjects studied

Chapter 4:

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Measures of central tendency: averages or measures of location that find a single number that reflects the middle of the distribution of scores – the ‘average’ / ‘mean’ score for that group

(Arithmetic) mean: often known as the ‘average’. ( (x- )=0 Summation symbol ( x): stands for summing/adding up all scores (‘x’) X-bar ( ): the mean value of the variable ‘x’ (when x is divided by ‘n’) Takes the distance between the mean and each score into consideration (one

extreme value increases mean)

Median : a value of the variable (‘x’) in which there are as many scores greater than the median as there are scores less than the median (the middle of the frequencies)

Median position: the person, place, or thing that possesses the median score (middle position)

Median position: n+1 2

Array: a listing from highest to lowest (or lowest to highest) Does not take the distance between the mean and each score into consideration (one extreme

value does not increase/influence median)

Mode: a category of a variable that contains more cases than can be found in either category adjacent to it – the category with the largest frequency (= the values and NOT their respective frequencies)

Modal class/modal category: where data have been grouped, a class interval/category that contains more cases than can be found in either category adjacent to it.

Requires a fairly large n for all modes to be correctly identified Continuous variable: a variable that is not limited to a finite number of scores (e.g. scores

with decimals) Unimodal: a distribution with one modes Bimodal: a distribution with two modes Trimodal: a distribution with three modes Modality: the number of modes found in the frequency distribution x-axis: the horizontal line of a graph f-axis: the vertical line of a graph Origin: point where the x-axis intersects the f-axis

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Frequency polygon: connection of the dots formed by drawing a straight line from each dot to the next dot, as x increases

Histogram: graph in which bars are created from one-half unit below each value of x to one-half unit above that value. The f-axis indicates the frequency of each score’s occurrence

Smooth curve: connection of dots (similar to a frequency polygon) but generated by a curved line fitting through the dots instead of a series of straight lines.

Major Formulas:

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Individual data Frequency distributionsThe Mean: The Mean:

= f x = fx n f

Median position=n + 1

2

Median position= n + 1 f + 1

2 = 2

Chapter 5:

Measures of dispersion/ variability: measures concerning the degree that the scores under study are dispersed or spread around the mean

Dispersion: the extent of clustering or spread of the scores about the mean

Range: the simplest measure of dispersion that compares the highest score and the lowest score for a given set of scores – insensitive to other scores other than lowest and highest

Mean deviation (M.D.): an average distance that a score deviates from the mean (found by summing all differences from the mean, and dividing them by the number of scores in the set)

Absolute value: the distance or difference, disregarding its sign / treating all differences as if they were positive numbers

Variance: an ‘average’ or mean value of the squared deviations of the scores from the means – distance from the mean

Variance =

Standard deviation: the positive square root of the variance, which provides a measure of dispersion closer in size to the mean deviation

Standard deviation=

Lev

els o

f m

easu

rem

ent Central Tendency Dispersion

Nominal Mode ---Ordinal Median ---

Ratio Mean Variance Interval Mean Standard Deviation

Types of statistics:

1.

6

M.D

Descriptive statistics: describing data obtained/ describing the sample / describing frequency distributions or relationships between variables

Inferential statistics: making statements based on what is found in samples/concerning the population

2.

7

Chapter 6

Contingency table/Cross-tabs: a table that depicts a possible relationship between the independent and the dependent variable (used for nominal & ordinal levels of measurement)

Dependent variables: placed in the table’s rowsIndependent variables: placed in the table’s columns

Calculating expected frequencies (no relation between variables)0.538 x 100 = 53.8 0.538 x 160 = 86.10.462 x 100 = 46.2

(Table data are often transformed into percentages, to make it easier to observe – we always compare percentages horizontally when calculating the percentages of the column totals)

When observing cross-tabs, we always:1) put the independent variable in the columns (i.e. its categories make up the columns)

2) calculate percentages of columns (as done in example)

3) compare percentages horizontally within the categories of the dependent variable

Control variable: a third variable that may have an influence on the relationship between the first two variables (equivalent to control group in experiments) – Two general outcomes when the control variable is entered:

1. The control variable has no impact on the initial relationship2. The presence of the control variable changes the initial relationship or is necessary for there to

be a relationship between the independent and dependent variables

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gender

likes poetry

male female total

likes A60(60%)

B80(50%)

140

(53,8%)

dislikes C40(40%)

D80(50%)

120

(46,2%)

total 100

100%

160

100%

260

53.8 (a) 86.1 (b)

46.2 (c) 74.02 (d)

Spurious relationship: relationship between two variables that is the product of a common independent variable (tested by looking for possible third ‘control’ variable)

Causal Models: Schematic diagrams showing the independent, dependent, and control variables and, where appropriate, positing the flow of causation of change in the dependent variable.

I: Independent variable p 174 (!)D: Dependent variableC: Control variable

The Independent and Control variable do affect the dependent variable – I and C are concurrent

The control variable has no impact on the dependent variable

or independent variable, but the independent variable does have an impact – C is irrelevant to the I-D relationship

The independent variable has no impact on the dependent

variable or the control variable, but the control variable does

have an impact on the dependent variable – I is irrelevant to the C-D relationship

The independent variable causes the control variable, which in turn affects the dependent variable – C is intervening

Antecedent variable: the variable initially leading to change in the dependent variable Intervening variable: the variable through which the antecedent variable brings about change in the dependent variable For example: Gender determines the type of values to which one is socialised. Gender (I) leads to values (D) through socialisation (C)Gender: antecedent variableSocialisation: intervening variable

Chapter 7:

Tests of statistical significance: techniques that help us to generalise to a larger group

Inferential/inductive statistics: the body of knowledge that deals with tests of significance

Population (sampling universe): the group about which we want to generalise

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I

D

C

I

D

CCI

D

CC

I

D

C

Variable 1 Variable 2 (= reciprocal causation between variables)

Sample: the smaller group from the population that is selected to be studied

Drawing a sample: the selection of subjects to be in a sample

Sampling bias/biased sample: the mechanism for selecting the sample that causes the sample to be not representative of the population as a whole

Random sample: a sample that is not biased

Simple random sample: a sample drawn in such a way that every member of the population has an equal likelihood of being included in the sample

Sampling error: a deviation from what actually exists in the population not associated with sampling bias but still existing, even though the sample was randomly drawn - we always have to assume a certain degree of sampling error (inherent in sampling)

- inferential statistics gives us the tools to deal with this uncertainty

Null hypothesis (H0): - There is no relationship between the variables in the population - There is no difference between the groups in the population

Alternative/research hypothesis: statement that indicates that in the population, the means of two or more groups differ or, in the case of a crosstab, the two variables are related

Fallacy of affirming the consequent: a principle of logic that suggests that the only way to ‘prove’ the alternative hypothesis is to demonstrate that the null hypothesis is untrue

Chi-square test: a test of significance for two variables in a crosstab – when crosstab is used – the test of significance that is acquired is always the Chi-square test.

Sample statistics: information computed from sample data

Population parameters: information computed from population data

Sample Population

Mean

Variance s2

Standard Deviation sSize n N

One-sample tests: rests that compare data from a sample to similar data in a population (to see if H0

can be rejected)

One-sample z tests: a test of significance that can be performed when we know the population’s standard deviation as well as its mean – uses z-table of z-values

One-sample t tests: a test that can be performed when we know the population’s mean but not its standard deviation – uses t-table of t-values

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Sigma hat: an estimate of sigma =

- When running a one-sample t-test in SPSS, it will not tell us directly whether or not to reject H0

- Instead, SPSS gives us a „p-value“: the probability of obtaining a difference (between the sample mean and the value we tested it against) at least as large as the one that was obtained (under the assumption that H0 is true)

- in other words: the p-value is the probability of falsely rejecting H0- In order to decide whether or not to reject H0, we compare the p-value with our alpha

(level of significance)- if p-value > alpha, do not reject H0- if p-value ≤ alpha, reject H0 & accept H1- Thus: low p-values are „better“ (better chance of rejecting H0) than larger p values- (Note: SPSS always assumes a nondirectional H1 and thus gives us the two-tailed

probabilities (p-values) – If our H1 is directional (one-tailed), we divide the p-value by two before comparing it to our alpha

Two-sample t test: a t test that compares two sample means, rather than one sample’s mean to another population’s mean

One-way ANOVA – f-test (analysis of variance): a test in which two or more sample means may be compared simultaneously

0.05 level of significance: the probability level classically used by statisticians for determining that a null hypothesis may be rejected

Probabilities: proportions that reflect the likelihood of a particular outcome occurring

Type 1 Error/Alpha error: the probability of falsely rejecting a true null hypothesisType 2 Error/Beta error: the probability that the null hypothesis is really false – H1 is true – but our obtained statistic – z,t, etc. – was too low to enable us to reject the H0, even though it ought to be rejected.

Non-directional alternative hypothesis: an alternative hypothesis that does not specify the directionality (which mean will ultimately be larger) 1 ≠ 2

Directional alternative hypothesis: an alternative hypothesis that does specify which mean will be the larger one 1 > 2 (or vice versa)

Degrees of Freedom: an additional piece of information needed for tests where critical values vary with the problem any may be functions of such things as sample size

X-Bar-Bar: The letter X with a double bar over it, used to represent the mean of all of the sample means of a population distribution.

Chapter 8

The Normal distribution: a family of frequency distributions - in inferential statistics, we often assume our variables to be „normally distributed“, meaning:

- unimodal - symmetric

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- asymptotic to x-axis

Standard score: a score designated by z, in which that score is expressed in standard deviation units from the mean

Sampling distribution of sample means: the frequency distribution that would be obtained from calculating the means of all theoretically possible samples of a designated size that could be drawn from a given population

Normality assumption: the assumption that the population being studied is normally distributed along variable x

Interval estimation: an interval of scores that is established within which a population’s mean is likely to fall, when that parameter is being estimated from sample data

Confidence interval: an estimated interval within which we are ‘confident’ – based on sampling theory – that the parameter we are trying to estimate will fall (e.g. if we want to be 95% „confident“, we use the critical values of z for the 0.05 level (two-tailed, 2.5% on the left & 2.5% on the right - thus looking for 0.025 in column C)

- we use the distribution of sample means to find the boundaries of the confidence interval

- E.g. if we want to be 95% „confident“, we use the critical values of z for the 0.05 level (two-tailed, thus looking for 0.025 in column C): 1.96

- Meaning: 95% of all sample means fall between μ - 1.96 SE and μ + 1.96 SE- We can then conclude with 95% confidence that μ lies between x - 1.96 SE and x +

1.96 SE

Chapter 9

The null-hypothesis and decision making:

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- moving from descriptive to inferential statistics means having to deal with uncertainty- we cannot prove our research hypotheses directly, we can only show that the null

hypothesis (H0) is very unlikely- how unlikely is unlikely enough to reject H0? What threshold to use?- we call this threshold the “level of significance” (alpha) - in social sciences, we typically use a 0.05 level of significance (alpha of 5%)- meaning: we make sure that the probability of falsely rejecting a true null hypothesis

is not greater than 5%- if H0 is true (no relationship in population between gender & poetry liking), we would

expect the two tables to be similar (of expected and observed frequencies)- however, even if H0 is true, we might still find a relationship in our sample (due to

sampling error)- thus, there is a chance that we wrongfully reject H0, i.e. assume that there is a

relationship in the population while there isnt (Type I error / alpha error)- the level of significance (alpha), defines the level at which we may reject H0, thus

setting the maximum probability of making a Type I error

How to decide whether the difference we found should be attributed to actual differences in the population?

- formulate H0 and H1 (e.g.: H0: H1:

- (remember: hypotheses are always about populations)- calculate a “test statistic” (here: one sample z-test)- compare the obtained value to a critical value (depends on level of significance)- if obtained value is equal to or exceeds the critical value, we may reject H0 (and

accept H1)

Standard error: the standard deviation of the distribution of sample means – designated with

Given a population with a mean µ and a standard deviation σ (normally distributed):- if we repeatedly take samples (size n) from that population and calculate the mean of

each sample, some sample means will be slightly higher than µ, some will be slightly lower than µ, and few would deviate greatly from µ (due to sampling error)

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- if we take all theoretically possible samples (size n) we can construct a new distribution of all those sample means: the distribution of sample means, which is normally distributed

- (NOTE: we never actually get to see this distribution, it is a theoretical construct!)- the mean of the distribution of sample means ( ) equals µ (put differently: the mean

of all possible sample means is equal to the population mean)- the distribution of sample means has a SD of , which is called the standard error

(SE) Why useful?

- since the distribution of sample means is normally distributed, we can use the z table to make statements about proportions/probabilities (similar to what we did last week)

- however, we are now working with a distribution of sample means, which has a different SD than the population distribution

- the formula we used last week for calculating z was:

- when we reformulate this formula for the distribution of sample means (i.e. replace the SD with the SE), we get:

(this is the formula of the one-sample z-test that we used earlier as a test statistic)

Two sample t-test: a comparison of two sample means – in order to generalise about a difference between the two population means. – H1 may be either directional or non-directional (making it directional means splitting up alpha to 2.5% to the left, 2.5% to the right ).

Steps in significance testing:1. formulate H0 & H1 (directional or nondirectional H1?)2. calculate the test statistic (in this case: z)3. find appropriate critical value of test statistic at the 0.05 (alpha) level (directional or

nondirectional H1?)4. compare obtained statistic with critical value5. if (absolute value of) obtained statistic is less than critical value, do not reject H0: no

statistical significance6. if (absolute value) of obtained statistic is equal to or larger than critical value, reject

H0, accept H17. state your conclusion based on H1

Independent two sample t-test: the composition of one sample is in no way matched or paired to the comparison of the other sample – the two samples reflect two separate populations

Dependent two-sample t-test: the t test used when the two samples are dependent samples (members of one sample are not selected independently, but are instead determined by the makeup of the other sample – matched pairs situation) – not focused on in this coursePaired difference t-test: a one sample t-test applied to the difference in each pair of scores (for dependent samples)

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Pooled estimate of common variance: estimate based on a weighted average of two sample variances being used to estimate the population variance in finding the SE.

F test for homogeneity of variances: a test, based on the sample variances, used to determine the most appropriate t test formula to use (either 1

2 = 12 or 1

2 ≠ 12)

Steps for the two-sample t test (independent samples):

1. write out H0 and H1 for the original problem, the comparison of the two sample means(H0: μ1 = μ2 H1: μ1 > μ2)

2. for each sample, determine its n, its and its variance, s2

3. to determine which t formula to use (that for equal population variances or that for unequal population variances), do the F tests for homogeneity of variances.

a. Write out H0 and H1 for the F-test (not the same as the ones for the t-test)(H0: 1

2 = 12 H1: 1

2 ≠ 12)

b. Calculate F and its two degrees of freedomc. Compare the obtained F to Fcritical, 0.05 level, taken from a table of critical

values of F.d. If Fobtained ≥ Fcritical, assume unequal population variancese. If Fobtained < Fcritical, asummer equal population variances

4. for the F test for homogeneity of variances, SPSS calculates a p-value (the chance of falsely rejecting H0, i.e. variances being equal)

5. in order to decide whether or not to reject H0 (equal variances), we compare the p-value with an alpha of 5%

6. if p-value > alpha, do not reject H07. if p-value ≤ alpha, reject H0 & accept H18. once we know whether to assume equal variances in the population, we can look at

the result of the t-test and (based on its p-value) reject or accept its H0 (i.e. the H0 of the t-test, which is about population means)

Tests of significance:

Compare sample mean to given value (e.g. population mean): does the sample mean differ significantly from the given value?

Based on our sample, can we assume that students spend more than 20min per day on the Internet?

H0: μstudents = 20 H1: μstudents > 20 (note: in this example, we formulate a directional alternative hypothesis)

à one sample t-test (one sample z-test if we know σ)

Need to:- Understand the idea behind the test- Interpret SPSS output- Calculate the test by hand (use table)

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Compare two sample means with each other: do the two sample means differ significantly from one another?

Based on our samples of art and economy students, can we assume the two groups to differ in their level of happiness?

H0: μart students = μeconomy students H1: μart students ≠ μeconomy students

(note: in this example, we formulate a nondirectional alternative hypothesis)

à two sample t-test (specifically: independent sample t-test)

Need to:- Understand the idea behind the test- Interpret SPSS output

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Look at the relationship between two variables in a crosstab (ordinal/nominal): is there a significant relationship between the two variables?

Based on our sample, can we assume that there is a relationship between religion and political party preference?

H0: in the population, there is no relationship H1: in the population, there is a relationship

à Chi Square test

Need to:- Understand the idea behind the test- Interpret SPSS output- Calculate the test by hand (use table)

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Compare two or more sample means: does at least one of the groups differ significantly from the other groups?

Based on samples of Germans, Dutch, French and Swedish, can we assume that the three groups differ in the amount of beer they drink during an average weekday?

H0: μGermans = μDutch = μFrench = μSwedish H1: at least one group differs significantly (here: H1 is always nondirectional)

à Analysis of Variance (F-test)

Need to:- Understand the idea behind the test- Interpret SPSS output

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In order to test hypotheses (about populations), we need to run the appropriate test of significance:

- one sample z-test H0: the population mean has a given value- one sample t-test (in situations where we have to estimate population variance)

same H0 as one sample z-test- two sample t-test (more specifically: independent samples t-test)

H0: the two group means are equal in the population- Chi Square test (for crosstabs) H0: there is no relationship between the two

variables in the population- F test (ANOVA) H0: all the group means are equal in the population

Chi-square Test:- test of significance used for determining statistical significance of the relationship

between two variables in crosstabs (ordinal / nominal)- based on difference between observed (fo) and expected (fo) frequencies (à week 2) - H0: there is NO relationship between the two variables in the population - formula (used in week 2 to calculate Cramer‘s V)

based on difference between fo and fe tables - the more fo values differ from fe values, the larger the value of Chi Square (see

formula), the less likely that H0 is true- again: steps in signifcance testing:- formulate H0 and H1 - note that for the Chi Square test, H1 is always nondirectional - calculate test statistic - in order to calculate Chi Square, we need to construct a fe table - find appropriate critical value (given alpha, and df)- for this Chi Square test, the df formula is:

- compare obtained and critical value and decide whether to reject H0

Statistical power: the likelihood that our test will reject the null hypothesis, when, in fact, H1, really is true.

Small effects: Hypothesised 0.2 σ difference between μ for the experimental Group and μ for the control groupsMedium effects: Hypothesised 0.5 σ difference between μ the population meansLarge effects: Hypothesised 0.8 σ population mean difference.

Chapter 12

One-way analysis of variance (ANOVA): when comparing two or more group means, we use a statistical technique called Analysis of Variance (ANOVA), with a test of significance called F-test

- it is an „extension“ of the independent samples t-test that can be used for more than two groups

- the H0 for the F-test states that in the population all group means are equal

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- the H1 states that in the population at least one group mean differs from the other group means: the H1 is always nondirectional (as opposed to H1 for independent samples t-test)

- once we have rejected H0, we need a post-hoc test that tells us which group(s) differ significantly from the rest

- with the F-test, we investigate to what extent the categories of one variable (e.g. nationality) help us to explain the variation of another variable (e.g. happiness)

- we use the Analysis of Variance (F-test) when comparing the mean of two or more groups

- often (but not exclusively) used in experimental research- the H0 of the F-test states that all population group means are equal:

H0: µ1 = µ2 = µ3 = … = µk (k being the number of groups)- the H1 states that at least two population group means differ- note: the H1 of the F-test is always non-directional- even if we can reject H0, we still do not know which group(s) differ (à posthoc test)

In Analysis of Variance, we distinguish between three different types of variances:

the total variance: how much x varies around the grand mean (of x1, x2, x3) the variance between groups: how much group means vary around the grand mean the variance within groups: how much x varies around the respective group means

As with any test of significance, we (SPSS) calculate a test statistic, in this case F- Remember that as the value of the test statistic increases, the likelihood of H0 being

true decreases, i.e. larger value indicate less probable H0’s.- The value of F is dependent on the size of the different types of variances so that:

if variances between groups get bigger (bigger group mean differences), F gets larger

if variances within groups get bigger (bigger within group variances), F gets smaller

Post-hoc test:- A post-hoc test (amongst others): the Scheffé’s test- similar to two-samples t-test, it looks at differences between two group means

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- for each pair of groups the H0 of the Scheffé’s test states that their population means are equal

- Once we reject H0, we need a post-hoc test (e.g. Scheffé) to see which groups differ significantly and in which manner

Approach to F:

the between group variance is expressed by way of the between groups-mean square: MSB

the within groups variance is expressed by way of the within groups-mean square: MSW

The value of F is then defined as:

meaning: the value of F gets larger (making it easier to reject H0) in situations where group means differ more (i.e., large between group variance) the groups are more “homogeneous” (i.e. less within group variance) with the F-test, we investigate to what extent the categories of one variable (e.g. nationality)

help us to explain the variation of another variable (e.g. happiness) The value of the F test is dependent on the variance between group means and the variance

within groups, in such a way that it becomes easier to reject H0: if group means differ largely (large between-groups mean square) if variances within groups are small (small within-groups mean square)

One-way & Two-way ANOVA:

so far, we have only dealt with situations with one independent variable (countries) however, it is also possible to include two independent variables (e.g. country & living in big

city Vs. rural area) by running a two-way ANOVA when including more than one independent variable, we sometimes find interaction effects:

in such a case, the effect of one independent variable depends on the value of the other independent variable

(e.g., we could have found that the effect of country on happiness is more pronounced for people who live in the rural area and much less visible for those living in the big cities.)

ANOVA (F-test) – lecture notes:

When one group mean differs from the others, it is already possible to reject the null hypothesis

Always a non-directional test No pre-testing, only post-testing is used for F-tests Grand mean: mean of all subjects (general mean for all groups – black dotted line in graph) The more the group means differ from the grand mean, the easier it is to reject the null

hypothesis Two way ANOVA: when two independent variables are included (e.g. country and living in

big cities vs. rural area) – creates the finding of interaction effect: the effect of one independent variable depends on the value of the other independent variable

Sum of squares: the sum of the squared deviations of the values of x from the mean

Mean square: the mean squared deviation of a score (value of x) from the mean of all scores

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Total sum of squares: the total of the squared deviations of scores about the grand mean

Between-groups sum of squares: the portion of the total sum of squares that can be accounted for by the variations of the category means about the grand mean

Within-groups sum of squares/error sum of squares: the portion of the total sum of squares left unexplained by the variations of the category means about the grand mean

Between-groups mean square: a variance estimate based on the between-groups sum of squares (MSB)

Between-groups degrees of freedom: the degrees of freedom based on the number of groups studied or categories of the independent variable

Within-groups mean square: a population variance estimate based on what the categories of the independent variable do not explain – the variation of scores within the groups (MSW)

Within-groups degrees of freedom: that portion of the total degrees of freedom not accounted for by the number of groups studied.

Robust: a term for F meaning that it is accurate even when underlying assumptions (such as equal population variances) are violated.

Chapter 12

Chi-Square test: tests the statistical significance of a relationship in a crosstab ,𝑥-2.=,--,(,f-o− .,f-e .,)-2.-,f-e...df= (# rows – 1) x (# columns – 1) x ,f-e.Chapter 13

Correlation coefficient/ measure of association: measure of strength of a relationship in which data are not grouped in table but are individual raw scores (Pearson’s r)

Pearson’s r: coefficient that is used when both variables are an interval or ratio level of measurement – based on covariance

To determine the strength of the correlation, we have to consider the absolute value (i.e., drop the minus, if there is one):

0-0.30: weak correlation 0.30 - 0.60: moderate correlation (e.g., r = - 0.5 indicates a moderate negative correlation) 0.60 – 0.80: strong correlation 0.80 – 1.0 : very strong correlation

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Coefficient of determination: indicates the proportion of variation in the dependent variable (y) that can be explained by variation in the independent variable (x)

Regression equation: the mechanism for estimating a y score from the respective x score

Correlation-regression analysis: the presentation of correlation and regression techniques together

Cramer’s V: a measure of association for crosstabs (nominal/ordinal level data) – produces a single number which allows us to determine the strength of the relationship between two variables

)1(

2

knV

N= number of casesk= either number of rows or column in the table, whichever is smallest

Fo= observed frequencyfe= expected frequency

fe= expected frequency =,𝑟𝑜𝑤 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑡𝑜𝑡𝑎𝑙 𝑥 𝑐𝑜𝑙𝑢𝑚𝑛 𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑡𝑜𝑡𝑎𝑙-grand total.Interpretation of Cramer‘s V:

<0.10: very weak / no relationship 0.10 - 0.20: weak relationship 0.20 – 0.30: moderate relationship > 0.30: strong / very strong relationship

Ordered pair: a set of two numbers in parentheses separated by a comma (e.g. 2,0 – x,y)

Exam Questions:

1. A researcher is interested in the relationship between gender and media preferences. Based on a random sample (n=100), he has constructed the crosstab provided below (gender & liking of Greys Anatomy [on 3 point scale])

a) Calculate the relevant percentages. Based on these percentages, what are your preliminary conclusions about the relationship between the two variables?b) Calculate Cramer's V. What are your conclusions based on the value of Cramer's V?c) Which test of significance is appropriate in this situation? Calculate the appropriate test (start by formulating H0).d) Based on c), what can you conclude about the relationship between gender and liking of Greys Anatomy?

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2. Assume a given population in which age is normally distributed with a mean of 30 and a standard deviation of 5.

a) When randomly selecting one subject, what is the chance of him/her being older than 50?b) When randomly selecting one subject, what is the chance of him/her being older than 20?c) When randomly selecting one subject, what is the chance of him/her being older than 35, but younger than 50?

3. Assume that a researcher has drawn a random sample of communication students. She has reason to believe that communication students spend more than 90mins per day on the internet. In her sample (n = 100), she finds a mean of 110 and a standard deviation of 10. She estimates the population standard deviation (sigma hat) to be 11.Based on these data, can we assume her to be right to assume that communication students spend more than 90mins on the Internet?

a) Which test of significance is needed here? Explain.b) Conduct the appropriate test. Start by formulating H0 and H1. Explain each step of the test.c) What are your conclusions based on your results?

4. Draw a scatterplot for the datamatrix provided below (10 cases, 2 variables). What can you say about the relationship between the two variables based on the scatterplot?

5. Calculate Pearson's r. What can you state based on the value of Pearson's r. Were your initial conclusions confirmed?

6. True or false? - the t-test can be used for comparing more than two groups- Cramer's V can take on any value between -1 and 1- Pearson's r can take on any value between -1 and 1- for a given crosstab, the larger Chi Square the larger Cramer's V- in an ANOVA, the larger the F value, the less likely H0 to be true- the standard deviation can be used for ordinal variables- the modus can be used for ordinal variables- we can reject the H0 of a given t-test if we find alpha < p-value

7. Provided is the SPSS output of a two samples t-test below.a) What is the independent, what is the dependent variable?b) What can you conclude based on the output? Be sure to discuss all relevant steps in your interpretation of the output.

8. Provided is the SPSS output of an ANOVA below.a) What is the independent, what is the dependent variable?b) What can you conclude based on the output? Be sure to discuss all relevant steps in your interpretation of the output.

Regression Analysis

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• regression analysis most commonly used multivariate technique to test causal models in behavioural sciences (in psychology: ANOVA)

• used in situations where both the indep. variable(s) (from now on: IV) and the dep. variable (from now on: DV) are measured on interval/ratio level:

with happiness as DV, we might use income as IV (instead of nationality)

• some central questions addressed by regression analysis:

– how well can we predict the DV using the IV(s)?

– what effect does an one unit increase in one IV have on the DV?

– how confident can we be that our findings can be generalized (tests of significance)?

• two main types:

– simple regression: one IV, one DV

– multivariate regression: two or more IVs, one DV

Regression lines

in two-dimensional space (two axes), a line is defined by the following formula:

Y = b0 + b1 X

b0: the intercept

the point where the line crosses the y-axis (here: 10)

b1: the slope

how much does the value of Y increase (or decrease) with a one-unit increase on X (here: 0.5)

Regression analysis

• in simple regression (one IV), we try to predict Y (DV) with X (IV)

• we do this by trying to express Y as a linear transformation of X in the following (known) form: Y = b0 + b1 X

• graphically, this is done by placing a line in the scatterplot that best represents the dots (cases) in the sample

how well can we predict happiness (DV on y-axis!) with income (IV on x-axis!)?

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0 1000 2000 3000 4000 5000

Regression: graphically

• the closer the dots are to the line, the better is our model in predicting Y

• the degree of inaccuracy (error) of our model is calculated by taking all ‚prediction errors‘ (difference between predicted value and actual value on Y: red dotted lines), square them - to get rid of minus signs - and add up those squared differences

• this gives us the sum of squared residuals (SSR)

• but how did we (does SPSS) find the right regression line in the first place?

• of all possible lines that we could have drawn in the scatterplot, the regression line is the line that has the smallest SSR (i.e. the smallest amount of prediction errors)

• some data fit their regression lines better than others

• larger vertical distances between dots and

line, i.e. more errors

• line fits the data less well

• à larger SSR

• less errors

• line fits the data better

à smaller SSR

• from our intercept and slope we can state that

– with X=0, the model predicts life expectancy to be … (intercept)

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– an increase of 1X should bring about an … (slope)

A measure of goodness of fit: R2

• once we have the regression line, we will want to know how well our model fits the data

• for this, we look at a measure of goodness of fit: R²

• R² tells us what percentages of the variance of the DV can be explained by the IV(s)

• in other words: how useful is the model for predicting the DV?

• R² ranges between 0 (no prediction possible ) and 1 (DV is completely determined by IV)

• graphically: the closer the dots are to the line, the higher is R2

• in our example, 44.5% of the variance in life expectancy can be explained by the number of hospital beds

• thus, 55.5% of the variance cannot be accounted for

• we could build a more elaborate model, including other variables that affect life expectancy (multiple regression), but the basic logic of regression stays the same if we have more than one IV

Making inferential statements with regression analysis

• Until now, we have not yet tried to make any inferential statements

• Two types of tests of significance are provided as part of a regression analysis:

- a F-test to test the significance for the model as a whole

H0: In the population, R² is zero

In other words: In the population, the model is completely useless for predicting the DV

- a t-test for each individual IV

H0: In the population, the effect of the given IV is zero

(note that this t-test is not on averages, but on effects!)

regression analysis

– used to test causal models

– one or more indep. variables X [IV] (interval/ratio)

– one dep. variable Y [DV] (interval/ratio)

How well can we predict the DV using the IV(s)?

the regression line is defined by:

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b0: the intercept of the line [in SPSS: “constant”]

graphically: where the regression line crosses the Y axis

substantially: the predicted value of Y (DV) when X (IV) is zero

b1: the slope of the line

graphically: the steepness of the line

substantially: how much the predicted value of Y increases (or decreases) when X is increased by one unit

also called the unstandardized regression coefficient

Standardized regression coefficient (beta)

– used in order to make statements / compare the strength of effects

– standardized to take on values between -1 and 1 [interpretation of strength of effects analogous to Pearson’s r, but now making

causal statement]

R²: a measure of the goodness of fit

– how well does our model (the IVs) predict the DV?

– R²: the proportion of variance in the DV that can be explained by the IV(s)

– ranges between 0 [no prediction possible] and 1 [DV completely determined by IV(s)]

R2 revisited:

1) how useful is the mean as a predictor?

• look at differences between observed and predicted values of Y (red dotted lines)

• we square those differences to get the total sum of squares (SST)

• the SST tells us how useful the mean is as a predictor (i.e. what is the degree of prediction error?)

2) how useful is the regression line as a predictor?

• look at differences between observed and predicted values of Y (red dotted lines)

• we square those differences to get the sum of squared residuals (SSR)

• the SSR tells us how useful our regression model is as a predictor (i.e. what is the degree of prediction error?)

We were interested in the improvement in prediction when using our regression model instead of the mean

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• This improvement is expressed in the difference between SST and SSR and is called the model sum of squares (SSM):

• SSM = SST - SSR

SSM tells us how much better the regression model is (than the mean) in predicting Y, i.e. the reduction in inaccuracy

• in other words, it is the amount of variance in Y that can be explained by the model

• while SST is the total amount of variance in Y

The value of R² tells us what proportion of the total variance can be explained by the model

Thus, we express it as:

From simple to multiple regression analysis

• in multiple regression, we have more than one IV

• we thus have more than one

– standardized coefficient

– unstandardized coefficient

– t-test

• this also means that our regression equation gets slightly more complex:

Way to interpret:

intercept (“constant” in SPSS):

if all IVs are zero (zero age, zero trust, zero religiosity), we expect happiness to be 6.683

unstandardized coefficients:

• if age increases by one unit (one year), we expect happiness to decrease by 0.016, keeping the effect of trust and religiosity constant

• if trust increases by one unit, we expect happiness to increase by 0.220, keeping the effect of age and religiosity constant

• if religiosity increases by one unit, we expect happiness to increase by 0.041, keeping the effect of age and trust constant

Coefficients in multiple regression analysis:

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• in multiple regression, the coefficients describe the effect of IVs, if the effects of the other IVs are held constant

• in other words, when calculating the effect of one IV, the effect of all other IVs are “cancelled out”

• this means that when adding an IV to the model, the coefficients and their p-values of the IVs that are already in the model can change (and most likely will)

Interaction effects:

• when we have more than one IV, we might encounter (and test for) interaction effects

• we use this term if the effect that one IV has on the DV, is dependent on the level of another IV

• in multiple regression, we thus distinguish between main effect (effects of IVs on DV) and interaction effects

• In this example, we might expect both IVs to have a main effect on the DV

• Additionally, we might expect there to be an interaction effect:

the effect that playing a violent video game has on violent behavior, might be stronger for those who have a general aggressive attitude

In other words:

• if you are the aggressive type, playing a violent video game might add more to your aggressiveness (as it resonates with existing patterns of behavior)

• If you are the peace-loving type, playing a violent video game might only have a marginal effect

Thus, an interaction effect: the effect of one IV (video game) is dependent on the level of another IV (aggressive attitude)

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Assumptions:

1. Linearity: - The type of regression analysis we have discussed assumes linear relationships

between the IVs and the DV\- In cases of non-linear relationships, the model will not fit the data well (e.g. high

prediction errors, low R2) 2. Lack of multicollinearity:

- there should be no strong correlation between any two IVs - put roughly, no two IVs should correlate stronger than r = 0.80- diagnostics available in SPSS that test for multicollinearity (e.g. VIF)- multicollinearity can:

lead to higher p-values of t-tests, i.e. we would conclude that a given effect is not significant even though H0 should have been rejected (type II – error)

reduce the size of R2

lead to unreliable coefficients 3. Homoskedasticity:

- with homoskedasticity, we assume the variance of the residuals to be constant for all values of the IVs

- in other words: the accuracy of our predictions should not depend on the value of one or more IV

overview of Regression:

• we can use regression in situations where we want to test a causal model with one or more IVs and one DV, all measured on interval/ratio level

• R2 tells us the proportion of the variance in the DV that can be explained by the model

• R2 is dependent on the amount of total variance (SST) and the amount of variance that is explained by the model (SSM)

• the intercept (in SPSS: “constant”) tells us the predicted value of the DV when all IVs are zero

• the unstandardized coefficients tell us the change in the predicted value of the DV when the given IV increases by one unit, with the effect of the other IVs held constant

• the standardized coefficients are used to compare/make statements about the strength of effects (between -1 and 1)

• one F-test is run for the model as a whole (H0: in the population, the model is not useful in predicting the DV, R2 = 0)

• for each IV, one t-test is run (H0: in the population, the effect of that IV is zero)

• regression analysis is based on a number of assumption, whose violations can lead to false conclusions about the sample data and (even worse) about the population (e.g. type II errors)

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