Intro: “BASIC” STATSIntro: “BASIC” STATS

CPSY 501 Advanced stats requires successful CPSY 501 Advanced stats requires successful completion of a first course in psych stats (a completion of a first course in psych stats (a grade of C+ or above) as a prerequisite grade of C+ or above) as a prerequisite

That requirement allows students to build on that That requirement allows students to build on that experience to work toward familiarity with experience to work toward familiarity with ANOVA and multiple regression analysis as ANOVA and multiple regression analysis as applied in counselling psychologyapplied in counselling psychology

This review covers topics that would be covered This review covers topics that would be covered in prereq courses. Not all courses are identical, in prereq courses. Not all courses are identical, of course, and some students may have to of course, and some students may have to “catch up” in some areas “catch up” in some areas

Note & apologyNote & apology: there seem to be : there seem to be some formatting problems with some formatting problems with colours, belatedly noted…. MM colours, belatedly noted…. MM

Basic Stats: Conceptual Basic Stats: Conceptual HeartHeart

Research Questions:Research Questions: numbers vs. numbers vs. “data” “data”

Variables & “levels of measurement”Variables & “levels of measurement” Designs: Between & within “subjects” Designs: Between & within “subjects”

EX: EX: tt-tests -tests Theory & conceptual work: Theory & conceptual work:

description vs. inference, description vs. inference,

Uses of Uses of t t -tests, correlations, Χ-tests, correlations, Χ22, etc. , etc.

CorrelationCorrelation

a measure of the a measure of the linearlinear relationship relationship between variables between variables

How do we tell if variables have a How do we tell if variables have a linear relationship? linear relationship?

It’s connected to It’s connected to variancevariance (we are (we are interested whether the variables ‘co-interested whether the variables ‘co-vary’)vary’)

CorrelationCorrelation

Thus, when one variable deviates Thus, when one variable deviates from it’s mean – does the other from it’s mean – does the other variable of interest also deviate from variable of interest also deviate from it’s mean in a similar (or directly it’s mean in a similar (or directly opposite) way? opposite) way?

There are a number of ways There are a number of ways variables can be related:variables can be related: Positive RelationshipsPositive Relationships Negative RelationshipsNegative Relationships Not RelatedNot Related

CorrelationCorrelation

SPSS provides a variety of SPSS provides a variety of correlational procedures that correlational procedures that standardize the covariance (the standardize the covariance (the relationship between variables), and relationship between variables), and provide us with a value that lies provide us with a value that lies between -1 and +1between -1 and +1

This is called the ‘correlation This is called the ‘correlation coefficient’coefficient’

Pearson’s Product-Moment Pearson’s Product-Moment Correlation Coefficient [Correlation Coefficient [rr] ]

Requires parametric data (b/c it’s based upon Requires parametric data (b/c it’s based upon ‘average deviation from the mean’).‘average deviation from the mean’).

It is the default option in SPSS It is the default option in SPSS

Example: (ExamAnx.Sav)Example: (ExamAnx.Sav) A researcher has collected data on anxiety, time A researcher has collected data on anxiety, time

spent reviewing material, and exam spent reviewing material, and exam performance…performance… AnxietyAnxiety was measured before the exam was measured before the exam Exam performanceExam performance was assessed via the student’s was assessed via the student’s

percentage mark on the exampercentage mark on the exam Time Spent RevisingTime Spent Revising measured exam revision time measured exam revision time

Pearson’s Correlation Pearson’s Correlation CoefficientCoefficient

Correlations

1 -.441** .397**

.000 .000

103 103 103

-.441** 1 -.709**

.000 .000

103 103 103

.397** -.709** 1

.000 .000

103 103 103

Pearson Correlation

Sig. (1-tailed)

N

Pearson Correlation

Sig. (1-tailed)

N

Pearson Correlation

Sig. (1-tailed)

N

Exam performance (%)

Exam Anxiety

Time spent revising

Examperformance

(%) Exam AnxietyTime spent

revising

Correlation is significant at the 0.01 level (1-tailed).**.

Significance

Value

Name of Correlation Statistic

Each variable is perfectly correlated with itself

Spearman’s Rho [Spearman’s Rho [ρρ, or , or rrss] ]

a non-parametric statistic, and so can be used a non-parametric statistic, and so can be used when your data violate parametric assumptions when your data violate parametric assumptions and/or distributional assumptions. and/or distributional assumptions.

Spearman’s tests work by first ranking the data, Spearman’s tests work by first ranking the data, and then applying Pearson’s equation to those and then applying Pearson’s equation to those ranks.ranks.

Example (grades.sav)Example (grades.sav)-Is there a relationship between students’ grade on -Is there a relationship between students’ grade on

a national math exam (GCSE) and their grade in a a national math exam (GCSE) and their grade in a university stats course (STATS)?university stats course (STATS)?

-NB: STATS grades coded according to ‘letter’ (so, -NB: STATS grades coded according to ‘letter’ (so, A=1, B=2, C=3, et.c); GCSE coded similarly.A=1, B=2, C=3, et.c); GCSE coded similarly.

Spearman’s Rho [Spearman’s Rho [ρρ, or , or rrss] ] (Non-parametric (Non-parametric

Correlations)Correlations)

Correlations

1.000 .455*

. .011

25 25

.455* 1.000

.011 .

25 25

Correlation Coefficient

Sig. (1-tailed)

N

Correlation Coefficient

Sig. (1-tailed)

N

Statistics Grade

GCSE Maths Grade

Spearman's rho

StatisticsGrade

GCSE MathsGrade

Correlation is significant at the 0.05 level (1-tailed).*.

Name of Correlation Statistic Used

Sample Size

The correlation is positive

Chi-Square [Chi-Square [ΧΧ22] ]

Evaluates whether there is a relationship Evaluates whether there is a relationship between 2 categorical variablesbetween 2 categorical variables

The Pearson chi-square statistic tests The Pearson chi-square statistic tests whether the 2 variables are independent…whether the 2 variables are independent…if the significance is small enough (i.e., if the significance is small enough (i.e., conventionally, significance is less conventionally, significance is less than .05), we reject the hypothesis that than .05), we reject the hypothesis that the two variables are independent the two variables are independent (unrelated), and accept the hypothesis (unrelated), and accept the hypothesis that they are in some way realted.that they are in some way realted.

tt-Tests: -Tests: Comparing Two MeansComparing Two Means

Moving beyond correlational research…Moving beyond correlational research… We often want to look at the effect of We often want to look at the effect of

one variable on another by one variable on another by systematically changing some aspect systematically changing some aspect of that variableof that variable

That is, we want to manipulate one That is, we want to manipulate one variable to observe it’s effect on variable to observe it’s effect on another variable.another variable.

tt-Tests-Tests

Related/DependentRelated/Dependent tt-tests-tests A repeated measures experiment that has 2 A repeated measures experiment that has 2

conditions (levels of the IV)conditions (levels of the IV) the the same subjectssame subjects participate in both conditions participate in both conditions We expect that a person’s behaviour will be the We expect that a person’s behaviour will be the

same in both conditions (external factors – i.e., same in both conditions (external factors – i.e., age, gender, IQ, motivation – will not change from age, gender, IQ, motivation – will not change from Condition 1 to Condition 2.Condition 1 to Condition 2.

Experimental Manipulation: we do something Experimental Manipulation: we do something different in Condition 1 than what we do in different in Condition 1 than what we do in Condition 2 (so the only difference between Condition 2 (so the only difference between conditions is the manipulation the experimenter conditions is the manipulation the experimenter made)made)

tt-Tests-Tests

Independent samples Independent samples tt-tests-tests We still have 2 conditions (levels of the IV), We still have 2 conditions (levels of the IV),

but different subjects participate in each but different subjects participate in each condition.condition.

So, differences between the two group means So, differences between the two group means can possibly reflect:can possibly reflect: The manipulation (i.e., systematic variation) The manipulation (i.e., systematic variation) Differences between characteristics of the people Differences between characteristics of the people

allotted to each group (i.e., unsystematic variation)allotted to each group (i.e., unsystematic variation) Question: what is one way we can try to keep the Question: what is one way we can try to keep the

‘noise’ in an experiment to a minimum?‘noise’ in an experiment to a minimum?

tt-Tests-Tests

tt-tests work by identifying sources of -tests work by identifying sources of systematic and unsystematic systematic and unsystematic variation, and then comparing them.variation, and then comparing them.

The comparison lets us see whether The comparison lets us see whether the experiment created the experiment created considerablyconsiderably more variation than we would have more variation than we would have got if we had just tested the got if we had just tested the participants w/o the experimental participants w/o the experimental manipulation.manipulation.

Example: Dependent Example: Dependent samplessamples

tt-Tests (or “paired -Tests (or “paired samples”)samples”)

12 ‘spider phobes’ exposed to a 12 ‘spider phobes’ exposed to a picture of a spider (picture of a spider (picturepicture), and on ), and on a separate occasion, a real live a separate occasion, a real live tarantula (tarantula (realreal))

Their Their anxietyanxiety was measured at each was measured at each time (i.e., in each condition).time (i.e., in each condition).

Example: Dependent Example: Dependent samples samples

(paired) (paired) tt-Tests-Tests

Paired Samples Correlations

12 .545 .067

12 .545 .067

Picture of Spider& Real Spider

Pair1

Picture of Spider& Real Spider

Pair2

N Correlation Sig.

Paired Samples Statistics

40.0000 12 9.29320 2.68272

47.0000 12 11.02889 3.18377

40.0000 12 9.29320 2.68272

47.0000 12 11.02889 3.18377

Picture of Spider

Real Spider

Pair1

Picture of Spider

Real Spider

Pair2

Mean N Std. DeviationStd. Error

Mean

Example: paired Example: paired tt-Tests-Tests

Paired Samples Test

-7.00000 9.80723 2.83110 -13.23122 -.76878 -2.473 11 .031

-7.00000 9.80723 2.83110 -13.23122 -.76878 -2.473 11 .031

Picture of Spider- Real Spider

Pair1

Picture of Spider- Real Spider

Pair2

Mean Std. DeviationStd. Error

Mean Lower Upper

95% ConfidenceInterval of the

Difference

Paired Differences

t df Sig. (2-tailed)

Standard Deviation of the difference between the means

Standard error of the differences b/w subjects’ scores in each condition

Degrees of Freedom (in a repeated measures design, its N-1)

SPSS uses df to calculate the exact probability that the value of the ‘t’ obtained could occur by chance

The probably that ‘t’ occurred by chance is reflected here

Example: Independent Example: Independent samplessamples tt-Test-Test

Recall: Used in situations where there are 2 Recall: Used in situations where there are 2 experimental conditions – and different experimental conditions – and different participants are used in each conditionparticipants are used in each condition

Example: SpiderBG.savExample: SpiderBG.sav 12 spider phobes exposed to a picture of 12 spider phobes exposed to a picture of

a spider (picture); 12 spider phobes a spider (picture); 12 spider phobes exposed to a real-life tarantulaexposed to a real-life tarantula

Anxiety was measured in each conditionAnxiety was measured in each condition

Group Statistics

12 40.0000 9.29320 2.68272

12 47.0000 11.02889 3.18377

ConditionPicture

Real Spider

AnxietyN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

.782 .386 -1.681 22 .107 -7.00000 4.16333 -15.63422 1.63422

-1.681 21.385 .107 -7.00000 4.16333 -15.64864 1.64864

Equal variancesassumed

Equal variancesnot assumed

AnxietyF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

Summary Statistics for the 2 experimental conditions

Assumption of parametric tests (e.g., t-tests) is that variances in the experimental conditions are ‘roughly’ equal.

If Levene’s test is significant, the assumption of homogeneity of variances has been violated

(N1 + N2) - 2 = 22

Significance value of ‘t’: since 0.107 > .05 (criterion of significance) there is no significant difference between the means of the 2 samples