Research Methods 1: Statistics

Front 1

Population

Back 1

- Target group for inference
- Parameters are Numerical Characteristics
- Large, unobtainable, often hypothetical

Front 2

Sample

Back 2

- Sub group of population
- Statistics are Numerical Characteristics
- Obtainable, Small

Front 3

μ (Mu)

Back 3

Population Mean

Front 4

x bar

Back 4

Sample Mean

Front 5

Regression Equation:
x

Back 5

score on predictor variable

Front 6

Simple Random Sampling

Back 6

Independent Selection
REDUCES bias in generalizations

Front 7

Research

Back 7

Scientific Structured Solving Problem

Front 8

Regression Equation:
a

Back 8

y-intercept, value of y' when x=0

Front 9

7 Topics that all Inferential Statistics have in Common

Back 9

1) Use of Descriptive Statistics
2) Use of probability
3) Potential for estimation
4) Sampling Variability
5) Sampling Distributions
6) Use of a Theoretical Distribution
7) 2 Hypotheses, 2 Decisions, 2 Types of Errors

Front 10

Steps of Scientific Methods

Back 10

1) Encounter and identify problem
2) Formulate Hypotheses and Define Variables
3) Think through consequences of hypotheses
4) Design study, run it, collect data, compute statistic, test hypothesis
5) Draw Conclusions

Front 11

Independent Variable (IV)

Back 11

- Manipulated by researcher
- Researcher CHANGES values of the variable
- Comes first in time

Front 12

Characteristics of Regression

Back 12

1) linear only
2) generalize only for x values in your sample
3) y is different from y'; y=y'+e
4) error is e=y-y'

Front 13

Dependent Variable (DV)

Back 13

- Measured by researcher
- Follows IV

Front 14

Extraneous Variable

Back 14

- Should be controlled by researcher
- Competitors to IV
- Influence DV

Front 15

Best Fitting Line

Back 15

Stats b&a are computed so as to minimize the sum of e squared (Least Squares Principle)

Front 16

Random Assignment

Back 16

- Purpose: Control EV
- When: After random sampling; form groups out of entire sample

Front 17

Variable

Back 17

Entity that is free to take on different values

Front 18

Partition Total Spread

Back 18

- Total = Explained+NotExplained
- for both proportion of spread and amount of spread

Front 19

Ways to control Extraneous Variables

Back 19

1) Randomization of subjects to groups
2) Keep constant for all subjects
3) Include in design

Front 20

Predictor Variable

Back 20

Comes first in time, but not manipulated

Front 21

Probability

Back 21

Relative Frequency

Front 22

Criterion Variable

Back 22

Follows Predictor Variable

Front 23

Sample Space

Back 23

all possible outcomes of a research project

Front 24

Operational Definition

Back 24

Type of variable is assigned depending on how it is used in the study

Front 25

Types of Relationships

Back 25

Causal, Predictive

Front 26

Causal Relationships

Back 26

IV causes DV
Keys:
a) manipulation of IV
b) randomization of subjects to groups
c) replication - N>1 for each group

Front 27

Elementary Event

Back 27

any one data point in sample space

Front 28

Predictive Relationships

Back 28

PV predicts CV
Keys:
a) no manipulation
b) no randomization of subjects to groups
c) have replication

Front 29

Event

Back 29

any collection of Elementary Events

Front 30

Types of Research

Back 30

True Experiment
Observational Research

Front 31

True Experiment

Back 31

a) Manipulation of Variable
b) Randomization of subjects to groups
c) Replication

Front 32

Observational Research

Back 32

a) No manipulation of Variable
b) No randomization of subjects
c) Replication

Front 33

P(Elementary Event)

Back 33

1 / (total # in sample space)

Front 34

Quantitative Data

Back 34

Data has numeric value

Front 35

P(Event)

Back 35

(# in event)/(total #)

Front 36

Qualitative Data

Back 36

Data has numeric label

Front 37

Aspects of Data

Back 37

Middle - central tendency, location, center
Spread - variability, dispersion
Skewness - departure from symmetry
Kurtosis - peakedness relative to normal curve

Front 38

Measures of Middle

Back 38

mean, median, mode, T20

Front 39

Conditional Probability

Back 39

P(A|B)=(# in A and B together) / (# in B)

Front 40

Measures of Spread

Back 40

range, midrange, s*^2, s*, s^2, s

Front 41

Standard/Unit Normal Distribution

Back 41

Mu = 0
Sigma^2 = 1

Front 42

Characteristics of a good measure of spread

Back 42

1) Stat = 0 if spread is zero
2) As spread increases, stat increases
3) Stat measures just spread, not middle

Front 43

Midrange (MR)

Back 43

Upper Hinge - Lower Hinge
UH - LH

Front 44

Median Position (MP)

Back 44

(N + 1) / 2

Front 45

Sampling Distributions Purpose

Back 45

To get probabilities of stat to make inferences to get information necessary to estimate parameters

Front 46

Hinge Position (HP)

Back 46

([MP] + 1) / 2

Front 47

Whiskers

Back 47

Lines drawn from a hinge to an adjacent value

Front 48

Sampling Distributions Definition

Back 48

A distribution of a statistic could be formed by drawing all possible samples of a given size N from some population, computing the stat for each sample, and arranging these stats in a distribution

Front 49

s*^2
Sample Variance

Back 49

{Σ(x-xbar)^2} / N

Front 50

3 things to know about sampling distributions of x bar

Back 50

1) Mu of x bar = Mu
2) Sigma squared of x bar = sigma squared/N
3) Shape normal IF
a)Population is normal
ORb)N is large

Front 51

s*
Sample Standard Deviation

Back 51

√ s*^2

Front 52

s^2
Unbiased Variance Estimate

Back 52

{Σ(x-xbar)^2} / (N – 1)

Front 53

Central Limit Theorem

Back 53

Shape is normal if N is large

Front 54

s

Back 54

√s(squared)

Front 55

Outliers

Back 55

Any real data values outside whiskers

Front 56

Unbiased

Back 56

Mu of stat = desired parameter

Front 57

z-score

Back 57

aspect of data = relative position/standing

"something minus it mean divided by its standard deviation"

Front 58

characteristics of z-scores

Back 58

1) mean of a set = 0
2) variance of a set = 1
3) shape is the same as the shape of the distribution of the somethings

Front 59

Characteristics of Normal Distributions

Back 59

1) symmetric, continuous, theoretical, unimodal
2) bell-shaped
3) scores range from -infinity to +infinity
4) has 2 parameters (mu and sigma squared)

Front 60

Hypothesis testing

Back 60

the process of testing tentative guesses about relationships between variables and populations

Front 61

2 Keys for probability in N(0,1)

Back 61

1) distribution symmetric
2) total area/probability = 1

Front 62

test statistic

Back 62

a statistic used only for the purpose of testing hypotheses

Front 63

Correlation and Regression have in common . . .

Back 63

1) x,y pairs of scores
2) linear relationships

Front 64

Correlation

Back 64

stat = r
Purpose is to measure the degree of linear relationship

Front 65

Regression

Back 65

Purpose: to measure form of function of linear relationship
Prediction Equation: y'=bx+a

Front 66

Assumptions

Back 66

Conditions placed on a test statistic necessary for its valid use in hyp. testing

Front 67

Characteristics of r

Back 67

1) works with 2 variables, x&y
2) -1.005<=r<=1.00
3) measures only linear relationships
4) r(squared) = proportion of variability in y that is explained by x
5) r undefined if x or y has zero spread
6) r is demensionless

Front 68

Assumptions for z of x bar

Back 68

1) pop. of obs. normal
2) obs. are independent

Front 69

Population Correlation Coefficient's Impact on r

Back 69

- restriction of range
- combining data
- outliers

Front 70

Correlation does NOT imply . . .

Back 70

causation

Front 71

Regression Equation:
y'

Back 71

predicted score on y (criterion variable)

Front 72

Null Hypothesis

Back 72

H sub naught: The hypothesis we test (decision to reject or retain)

Front 73

Regression Equation:
b

Back 73

slope

Front 74

Alternative Hypothesis

Back 74

H sub one: where we put what we believe

Front 75

Significance level

Back 75

the standard for what we mean by a small probability in hypothesis testing, alpha = .05

Front 76

Directional hypothesis

Back 76

any Hypothesis with <,>,<=,>=

Front 77

Nondirectional hypotheses

Back 77

do not specify direction, eg not equal

Front 78

One-tailed Test

Back 78

uses only one tail of the sampling distribution of the test

Front 79

Critical values

Back 79

values of the test statistic that cut off alpha in the tail(s) of the sampling distribution

Front 80

Rejection Values

Back 80

valus of hte test stat for which we would reject Hnaught.

Front 81

Critical Value Decision Rules

Back 81

Reject null Hypothesis if the test stat is more extreme than a critical value

Front 82

p-value Decision Rule

Back 82

Reject Null if both are true:
1).5SAS p-value <= alpha
2) the result (test statistic) agrees with the alternative

Front 83

Type I Error

Back 83

Reject Null Hypothesis given Null is true

Front 84

Type II Error

Back 84

Retain Null Hypothesis given Alternative is true

Front 85

p(Type I Error)=

Back 85

p(rej. Null|Null true) = alpha

Front 86

p(Type II Error)=

Back 86

p(ret. Null|Alternative true) = Beta

Front 87

Effect size relationship to power

Back 87

as Effect size increase, power increases

Front 88

N (sample size) relationship to power

Back 88

as N increases, power increases

Front 89

Sigma squared relationship to power

Back 89

as sigma squared decreases, power increases

Front 90

alpha's relationship to power

Back 90

as alpha increases, power increases

Front 91

directional hypothesis influence on power

Back 91

gives best power if correct in predicting direction, but zero power if you are wrong

Front 92

non-directional hypothesis influence on power

Back 92

gives good power in both directions

Front 93

properties of t-distribution

Back 93

-A family of distributions
-Have one parameter = df
-Mu of t = 0
-sigma^2 of t =df/(df-2)]>1
-Symmetric,sort of bell-shaped

Front 94

t=

Back 94

(x bar-mu)/ sqroot(s^2/N)

Front 95

degrees of freedom

Back 95

-parameter of t-distribution
-(# of independent components - #of statistics)
-for 1-sample t, =N-1
-associated with s^2

Front 96

one-sample t assumptions

Back 96

-population bivariate normal
-subjects are independent

Front 97

Correlation

Back 97

-one sample
-x,y pairs
Null: rho=0

Front 98

Correlation statistic

Back 98

r

Front 99

Correlation degrees of freedom

Back 99

N-2

Front 100

2 ind sample t-test

Back 100

-2 independent samples
-Null: mu1=mu2
-Sigma^2 unknown
-Use when n1=n2>=15

Front 101

2 ind sample t-test degrees of freedom

Back 101

n1+n2-2

Front 102

2 independent sample t-test assumptions

Back 102

-Populations of observations are normal
-sigma1^2=sigma2^2
-observations are independent

Front 103

AWS t'

Back 103

-2 samples
-independent samples
-Null:mu1=mu2
-sigma^2 unknown
-use when n1=n2<15 or n1!=n2

Front 104

AWS t' test statistic

Back 104

(xbar1-xbar2)\
sqroot((s1^2\n1)+(s2^2\n2))

Front 105

AWS t' degrees of freedom

Back 105

n1+n2-2

Front 106

AWS t' assumptions

Back 106

-populations of observations are normal
-observations are independent

Front 107

2 dependent sample t

Back 107

-2 samples
-dependent samples
-x,x pairs
-sigma^2 unknown
-Null:Mu(sub d)=0

Front 108

2 dependent sample t statistic

Back 108

dbar-0/sqroot(s(sub-d)^2/N)

N=Number of pairs

Front 109

2 dependent sample t statistic degrees of freedom

Back 109

N-1

N=#of pairs

Front 110

2 dependent sample t Assumptions

Back 110

-population of d's is normal
-d's are independent

Front 111

3 Ways of acquiring x,x pairs for 2 dependent sample t-test

Back 111

1)researcher produced-researcher matches on extraneous variable
2)naturally occurring-come to researcher already paired
3)repeated measures-pre & past measurements

Front 112

Robustness assumptions

Back 112

1)normality-not met-robust
2)o1^2=o2^2-not met-robust if n1=n2>=15
3)independence-met-not robust

Front 113

Robustness Definitions

Back 113

1)the quality of a test stat when assumption is not met
2)the sampling distribution is well-fit by the theoretical distribution
3)alpha(true)~=alpha(set)=0.05: .04<=alpha(true)<=.06

Front 114

1-way ANOVA

Back 114

-for comparing multiple samples
-J=#groups, n=#obs, N=nJ
-Null:mu1=mu2=mu(sub-j)
-Alternative: any difference in mu(sub-j)s

Front 115

Logic of the Anova
Part 1

Back 115

1)We find 2 sample variances, one based on the x-bars, and one based on observations within groups: these 2-sample variances should estimate sigma^2 if Null is true, but estimate different quantities if alternative is true

Front 116

Logic of the Anova
Part 2

Back 116

2)We form an f-stat by putting the variance based on xbars in the numerator and variances based on observations in denominator.
If Null true, f~1.
If Alternative true, f>1.

Front 117

ANOVA based on xbar

Back 117

n*s(sub-xbar)^2
Null: estimates sigma^2
Alter:est. sigma^2+positive quantity

Front 118

ANOVA based on observations

Back 118

s(sub-pooled)^2=sum(s(sub-j)^2)/J
Null: est. sigma^2
Alternative: est. sigma^2

Front 119

ANOVA F

Back 119

F=n*s(sub-xbar)^2/
sum(s(sub-j)^2)/J
Null: expect F~1
Alternative: expect F>1

Front 120

s*^2 Sampling Distribution

Back 120

u=((N-1)/N)*sigma^2
shape: positively skewed

Front 121

r Sampling Distribution

Back 121

u=P if P=0
shape: symmetric, not normal

Front 122

s^2 Sampling Distribution

Back 122

u=sigma^2
shape: positively skewed

Front 123

MCP

Back 123

Multiple Comparison Procedure

Front 124

Pairwise Comparisons
For J means, there are ... pair wise comparisons

Back 124

C=J(J-1)/2

Front 125

MCP hypotheses

Back 125

Null: u(sub-j)=u(sub-j')
Alt: u(sub-j)!=u(sub-j')

Front 126

Error Rates for MCP
α'=alpha for each comp
c=# pair wise comparisons

Back 126

p()=p(at last one Type I error)
--goal to keep p() small
α'<=p()<=1-(1-α')^c<=c(α')

Front 127

Error Rate per comp. (gives good power)

Back 127

-Set α'=.05
-p() can be large

Front 128

Error rate family-wise

Back 128

Controls p() at α=.05 for c Comparisons by making α' small

Front 129

Tukey MCP situation/hypothesis

Back 129

J>=2 ind samples
H0: u(sub-j)=u(sub-j')
All p.w. comparisons
equal n's >=15

Front 130

Tukey AND Fisher-Hayter test stat

Back 130

t=xbar(sub-j)-xbar(sub-j')/
sqrt((MeanSquareWithin*2)/n)

Front 131

Fisher-Hayter MCP situation/hypothesis

Back 131

Overall ANOVA F must be significant
J>=2 ind samples
H0: u(sub-j)=u(sub-j')
All p.w. comparisons
equal n's >=15

Front 132

Tukey and Fisher-Hayter assumptions

Back 132

1.Pops. of obs. are normal
2.sigma(sub-j)^2=sigma(sub-j')^2
3.obs. are ind.

Front 133

Tukey MCP distribution

Back 133

q/sqrt(2)
J, degrees of freedom within, and alpha=.05

Front 134

Fisher-Hayter MCP

Back 134

q/sqrt(2)
J-1, degrees of freedom within, and alpha=.05

Front 135

Tukey MCP robustness

Back 135

very similar to 2 independent sample t

Front 136

2-Way ANOVA Situation/hypothesis

Back 136

2 Factors, J=# levels of A, K=# levels of B, n obs/cell, N=nJK
A)H0:u1=u2...=uJ
B)H0:u1=u2...=uK
AB)H0:No Interaction Effect

Front 137

2-Way ANOVA test stat

Back 137

FA=MSA/MSW=(SSA/dfA)/(SSW/dfW)
(FB and FAB, substitute B and AB for A, respectively)
dfA=J-1, dfW=JK(n-1)
dfB=K-1, dfAB=(J-1)(K-1)

Front 138

2-Way ANOVA distribution

Back 138

FA~F(sub-(J-1,JK(n-1)))
FB~F(sub-(K-1,JK(n-1)))
FAB~F(sub-((J-1)(K-1),JK(n-1)))

Front 139

2-Way ANOVA assumptions

Back 139

1.Pops of obs are normal
2.equal population variances for each cell
3.obs. are independent

Front 140

Factor

Back 140

a variable that classifies/groups subj.

Front 141

1-Way ANOVA situation/hypothesis

Back 141

1-factor
J>=2 independent groups
H0:u1=u2=uJ
H1:any difference in uJ

Front 142

1-Way ANOVA test statistic

Back 142

F=MSB/MSW=(SSB/dfB)/(SSW/dfw)
dfB=J-1 dfW=N-J=J(n-1)

Front 143

1-Way ANOVA distribution

Back 143

F~F(sub-(J-1),(N-J)

Front 144

1-Way ANOVA assumptions

Back 144

1.pops of obs are normal
2.sigma1^2=sigma2^2...=sigmaJ^2
3.obs are independent

Front 145

Levels

Back 145

a value of a factor (One in 1-Way Anova)