psyc512: research methods psyc512: research methods lecture 8 brian p. dyre university of idaho

PSYC512: Research MethodsPSYC512: Research Methods

PSYC512: Research MethodsPSYC512: Research MethodsLecture 8Lecture 8

Brian P. DyreBrian P. Dyre

University of IdahoUniversity of Idaho


Lecture 8 OutlineLecture 8 Outline

Questions about material covered in Questions about material covered in Lecture 7Lecture 7 Measures: Reliability, Precision, and Measures: Reliability, Precision, and

ValidityValidity Defining Variables and Research DesignsDefining Variables and Research Designs

Describing DataDescribing Data Testing HypothesesTesting Hypotheses Inferential StatisticsInferential Statistics


Understanding VariabilityUnderstanding Variability

What is variability?What is variability?

How is variability related to probability?How is variability related to probability?


Visualizing Variability: Visualizing Variability: Distributions of Frequency and Distributions of Frequency and the Histogramthe Histogram

Histograms: used to represent Histograms: used to represent frequencies of data in different frequencies of data in different classes or categoriesclasses or categories

BinBin FrequencyFrequency

00 00

11 00

22 00

33 00

44 33

55 11

66 66

77 44

88 22

99 66

1010 11


Displaying Histograms: Displaying Histograms: Stem and Leaf PlotsStem and Leaf Plots

Stem and Leaf plots are used to display histograms Stem and Leaf plots are used to display histograms graphically (on their side) using only typed graphically (on their side) using only typed characterscharacters

StemStem LeafLeaf (hypothetical histogram for (hypothetical histogram for IQ)IQ)66 787877 356683566888 01223444555566777788901223444555566777788999 0001123333333444556666788999900011233333334445566667889999

1010 0111223333444444556667777788889901112233334444445566677777888899 1111 00011222334445667778990001122233444566777899 1212 00125690012569 1313 0202


Distributions of Probability Distributions of Probability DensityDensity

Similar to frequency Similar to frequency histogram except y-histogram except y-axis now represents axis now represents probability density probability density (mass) rather than (mass) rather than frequencyfrequency

Probability density = Probability density = Frequency/NFrequency/N

Grade

0 1 2 3 4 5 6 7 8 9 10

Pro

babi

lity

Den

sity

0.0

0.1

0.2

0.3

0.4

0.5


Some Types of Probability Some Types of Probability Density DistributionsDensity Distributions

Normal (Gaussian)Normal (Gaussian)GammaGamma

Data

0 5 10 15 20 25

Pro

ba

bili

ty D

en

sity

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14


Describing Distributions:Describing Distributions:Estimators and ParametersEstimators and Parameters

Sample statistics Sample statistics estimateestimate population parameters, e.g. population parameters, e.g. Sample mean: Sample mean: MM or estimate the mean of a population, or estimate the mean of a population, Sample variance: Sample variance: ss22 estimates the variance of a population, estimates the variance of a population, 22

Properties of EstimatorsProperties of Estimators Sufficiency: extent to which statistic uses all information Sufficiency: extent to which statistic uses all information

(observations) available in sample(observations) available in sample Unbiasedness: extent to which expected value of statistic Unbiasedness: extent to which expected value of statistic

approaches population value with increased samplingapproaches population value with increased sampling Efficiency: tightness of cluster of sample statistics relative to Efficiency: tightness of cluster of sample statistics relative to

the population parameterthe population parameter Resistance: extent of influence of outliers on sample statisticResistance: extent of influence of outliers on sample statistic

X


Measures of the Center of a Measures of the Center of a Population or SamplePopulation or Sample

Measures of center represent the general magnitude of scores Measures of center represent the general magnitude of scores Mode: most frequent scoreMode: most frequent score Median: the middle score of an ordered listMedian: the middle score of an ordered list Mean (average):Mean (average): where where XX represents represents

a a vector of samples vector of samples andand

NN is the total is the total number number of of observations;observations;

Which measures are the most sufficient? Unbiased? Efficient? Which measures are the most sufficient? Unbiased? Efficient? Resistant?Resistant?

populationfor

asdabbreviateoften1

N

X

N

XX

N

ii


Measures of the Spread of a Measures of the Spread of a Population or SamplePopulation or Sample Measures of spread are used to assess the consistency of Measures of spread are used to assess the consistency of

scores in a distributionscores in a distribution Range = max score – min scoreRange = max score – min score Interquartile range = score(Q3) – score(Q1)Interquartile range = score(Q3) – score(Q1) Variance (Variance (22ss22) and standard deviation () and standard deviation (ss))

where where XX is a vector of data, is a vector of data,

is the mean of the is the mean of the population, is the mean population, is the mean of a sample, and of a sample, and NN is the is the total number of total number of observationsobservations

Which measures are the most sufficient? Unbiased? Which measures are the most sufficient? Unbiased? Efficient? Resistant?Efficient? Resistant?

N

X

2

2

1

2

2

N

XXs

X


Standard DeviationStandard Deviation

Standard Deviation (Standard Deviation () = sqrt(variance)) = sqrt(variance)

where X is the data, where X is the data,

m is the mean of the m is the mean of the data, data, and N is the total and N is the total number number of of observationsobservations

Remembering how to compute varianceRemembering how to compute variance

““the mean of the squares – square of the the mean of the squares – square of the means”means”

N

X

2

222

N

X

N

X


Describing Distributions Describing Distributions Parametrically: Statistical Parametrically: Statistical MomentsMoments

Any distribution based on interval or ratio data can Any distribution based on interval or ratio data can be summarized by its be summarized by its statistical momentsstatistical moments

First Moment: Mean—location of distribution on x-First Moment: Mean—location of distribution on x-axisaxis

Second Moment: Variance—dispersion of Second Moment: Variance—dispersion of distributiondistribution

Third Moment: Skewness—symmetry of Third Moment: Skewness—symmetry of distributiondistribution

Fourth Moment: Kurtosis—degree of “peakedness”Fourth Moment: Kurtosis—degree of “peakedness”


Testing HypothesesTesting Hypotheses

Hypothesis testing is the process by which Hypothesis testing is the process by which hypothetical relationships between hypothetical relationships between intervening intervening variables are assessedvariables are assessed

Hypotheses are always tested relative to Hypotheses are always tested relative to one-another or to a “null” hypothesisone-another or to a “null” hypothesis

ExamplesExamples Comparing groupsComparing groups Assessing performance interventionsAssessing performance interventions Assessing relationships between Assessing relationships between

variablesvariables


Null-Hypothesis Testing Null-Hypothesis Testing and Inferential Statisticsand Inferential Statistics

2 possible realities2 possible realities Relationship between your variables does not Relationship between your variables does not

exist—a null relationship (exist—a null relationship (HHoo, the , the null null hypothesishypothesis))

Relationship between the two variables in Relationship between the two variables in question actually exists (question actually exists (HH11, the , the experimental experimental or alternative hypothesisor alternative hypothesis))

2 possible decisions when looking at the data2 possible decisions when looking at the data Conclude that a relationship exists (Conclude that a relationship exists (reject the reject the

null hypothesis, Hnull hypothesis, Ho o DISCONFIRMATION!) DISCONFIRMATION!) Conclude that no relationship exists (do not Conclude that no relationship exists (do not

reject the null hypothesis reject the null hypothesis CONFIRMATION? CONFIRMATION? NO!)NO!)


Null-Hypothesis Testing and Null-Hypothesis Testing and Inferential StatisticsInferential Statistics

HHo o TrueTrue HHo o FalseFalse

Reject Reject HHo o

(conclude there (conclude there is an effect)is an effect)

Type I Type I error error (false (false alarm)alarm)

Correct Correct DecisionDecision

Do not Do not Reject Reject HHo o

(conclude there (conclude there is NOT an is NOT an

effect)effect)

Correct Correct DecisionDecision

Type II Type II error error (miss)(miss)

Decision

True State of the World2 realities by 2 decisions form a 2 x 2 matrix of 4 possibilites


Hypothesis Testing: Hypothesis Testing: Probability and Probability and StatisticsStatistics Problem: How do we distinguish real differences or Problem: How do we distinguish real differences or

relationships from measurement noise?relationships from measurement noise?

Probability and statistics may be used to assess (descriptive Probability and statistics may be used to assess (descriptive statistics) or compare (inferential statistics) the relative statistics) or compare (inferential statistics) the relative magnitude of different types of variabilitymagnitude of different types of variability

Effect (treatment) VarianceEffect (treatment) Variance

Variability due to relationship between variables or Variability due to relationship between variables or effect of different levels of independent variable effect of different levels of independent variable (treatments)(treatments)

““Good” variance that we want to maximizeGood” variance that we want to maximize

Error VarianceError Variance

Variability in measure due to factors other than the Variability in measure due to factors other than the treatmenttreatment

““Bad” variance that we want to minimizeBad” variance that we want to minimize


Hypothesis Testing: Hypothesis Testing: Inferential StatisticsInferential Statistics

All All inferential statistics are evaluating this ratio:inferential statistics are evaluating this ratio:

Effect (good) VarianceEffect (good) VarianceTest statistic =Test statistic = -------------------------------------- --------------------------------------

Error (bad) VarianceError (bad) Variance

Example test statistics: Example test statistics: Chi-square, t, FChi-square, t, F These test statistics have known distributions that These test statistics have known distributions that

then allow us to estimate then allow us to estimate p, p, the probability of a the probability of a Type I error (inappropriately rejecting the null Type I error (inappropriately rejecting the null hypothesis)hypothesis)

Decision to reject null is made by comparing Decision to reject null is made by comparing pp to to some generally accepted criterion for Type I error some generally accepted criterion for Type I error probability, probability,



Why might we observe Why might we observe a difference between a difference between two groups if no two groups if no difference actually difference actually exists (null is true; exists (null is true; samples are drawn samples are drawn from the same from the same population)?population)?

Each sample may Each sample may have a unique mean have a unique mean due to sampling errordue to sampling error

Fre

quen

cy

2X

Fre

quen

cy

1 Population

2 samples

1X



How does this change How does this change if a difference actually if a difference actually exists between my exists between my groups?groups?

Each sample has a Each sample has a unique mean that unique mean that represents both represents both sampling error and the sampling error and the differences between differences between the 2 populationsthe 2 populations

Fre

quen

cy

2 Populations

Fre

quen

cy

2X1X


How is How is p p calculated? It depends on…calculated? It depends on…

the scaling properties of your dependent variable the scaling properties of your dependent variable (DV)(DV)

DV is interval or ratioDV is interval or ratio parametric tests parametric tests DV is nominal or ordinalDV is nominal or ordinal non-parametric tests non-parametric tests

Research designResearch design Experimental – test differences on measure between Experimental – test differences on measure between

conditions or groups conditions or groups t-test, ANOVA, sign test, t-test, ANOVA, sign test, Mann-WhitneyMann-Whitney

Correlational – test relations between different Correlational – test relations between different measures measures Pearson product-moment correlation, point- Pearson product-moment correlation, point-biserial correlation, etc.biserial correlation, etc.

the manner in which you phrase your hypothesesthe manner in which you phrase your hypotheses One tailed vs. two-tailed testsOne tailed vs. two-tailed tests


Next Time…Next Time…

Topic: Normality, Probability, Nuts and Topic: Normality, Probability, Nuts and Bolts of Testing HypothesesBolts of Testing Hypotheses

Be sure to:Be sure to: Read the assigned readings (Howell Read the assigned readings (Howell

chapters 6-7)chapters 6-7) Continue searching and reading the Continue searching and reading the

scientific literature for your proposalscientific literature for your proposal

psyc512: research methods psyc512: research methods lecture 8 brian p. dyre university of idaho

Documents

research methods psyc512

sample variance

sample mean

parameters sample statistics

research designs

population parameters

sample unbiasedness

sample sufficiency