@ProfAndyField

Making Statistics Easy by Getting your PENIS out in

the Classroom

Professor Andy Field

@ProfAndyField

Outline

!   Part 1: !   Why should students love statistics?

!   Part 2: !   Why do students hate statistics?

!   Part 3: !   Can we make them like statistics more by exposing

them to my PENIS?

@ProfAndyField

A Scientist (Arguably)

Intelligent Laypeople

@ProfAndyField

Statistics is a life skill ! Utts (2003) some core skills:

1.  When causal relationships can and cannot be inferred. 2.  The difference between statistical significance and

practical importance. 3.  The difference between finding ‘no effect' and finding

no statistically significant effect. 4.  Sources of bias in surveys and experiments, such as poor

wording of questions, volunteer response, and socially desirable answers.

5.  Understanding that variability is natural, and that ‘normal’ is not the same as ‘average’ (e.g., child development).

!   Gordon (2004) !   7% of Psychology students think statistics is generally

useful !   16% thought it was useful for psychologyJ

@ProfAndyField

How the contraceptive pill works

@ProfAndyField

The Daily Mail Said... !   “Women may suffer a permanent decline in sex drive after

taking the contraceptive pill, researchers have said.”

!   “A number of sexual dysfunction effects are associated with the Pill, including dulled libido ... Until now it has always been assumed that these are reversible, and cease to be a problem as soon as a woman comes off the Pill. But new research suggests that the effect on libido might be long lasting or even permanent.”

!   “A team of American researchers ... studied 125 young women attending a sexual dysfunction clinic. Sixty two were taking oral contraceptives, 40 had previously taken them, and 23 had never been on the Pill.”

!   “The scientists measured levels of SHBG in the women every three months for a year, and found they were seven times higher in users of the Pill than in women who had never taken them.”

!   “Levels declined in women who had stopped taking the Pill, but remained three to four times higher than they were in those with no history of using oral contraceptives.”

@ProfAndyField

The Layperson Might Think !   At Best:

-  SHBG levels were seven times higher in users of the Pill than in women who had never taken them (80%)

-  Although levels of SHBG in women who stopped taking the pill were 3-4 times higher than women who had never taken the pill, this is not a meaningful difference (33% agree)

!   At worst:

-  The OC causes permanent loss of Libido (33%)

-  The OC causes sexual dysfunction (50%)

-  Women taking the Pill are at risk from permanent loss of libido even when they stop taking it (83%)

-  GPs should inform women of this risk when prescribing the pill (100%)

Panzer et al.

0

50

100

150

200

250

Never Taken Pill

On Pill Stopped Taking Pill

Lib

ido

(SH

GB)

Group

Baseline

>120 Days

Normal

Lower

Panzer et al.

42 35

209

80

0

50

100

150

200

250

Baseline > 120 Days

Lib

ido

(SH

GB)

Group

Never Taken Pill

Stopped Taking Pill

Normal

Lower

@ProfAndyField

The Psychologist Copy Editor ...

!   “Women with sexual dysfunction may suffer a permanent (well, up to 3-6 months or a year) decline in levels of SHBG after taking the contraceptive pill, researchers have said.”

!   “A number of sexual dysfunction effects are associated with the Pill, including dulled libido ... Until now it has always been assumed that these are reversible and in the current research SHBG did actually decline a lot after coming off of the Pill, and cease to be a problem as soon as a woman comes off the Pill. But new research suggests that the effect on SHBG might be long lasting or even permanent even though we only have data over, on average 3-6 months.”

!   “The scientists measured levels of SHBG in the women every three months for a year (well, some of them, the average was 3-6 months), and found they were seven (in a parallel universe where 7 = 5) times higher (at baseline, not after they’d come off the pill) in users of the Pill than in women who had never taken them.”

!   “Levels declined in women who had stopped taking the Pill, but remained three to four times higher (in another parallel universe where 3 to 4 means 2.29) than they were in those with no history of using oral contraceptives. However, women in the discontinued group were followed up on average for 73 days less than never users (or 38 days less for the long term follow up group).”

!   Subjective libido was never compared at follow-up in discontinued users and never-users, so we don’t really know a lot about libido one way or another.

@ProfAndyField

A Zombie Quiz

Brain Chips!

Potato Chips!

H! 28! 42!

Z! 61! 57!

!   I collect data about how many humans and zombies choose brain chips or potato chips to accompany their dinner.

!   How do I analyze these data?

@ProfAndyField

A Chi-square Test?

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A Spearman Correlation?

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A Kendall’s Tau-b Correlation?

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A (Pearson) Correlation

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A t-Test?

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Linear Regression?

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Wilcoxon-Mann-Whitney Test?

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Kruskal-Wallis Test

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Logistic Regression

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Ordinal Logistic Regression (PLUM)

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One-Way ANOVA

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Linear Mixed Model

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Discriminant Function Analysis

GLM

One

Two or more

Continuous

Categorical

How many outcome variables?

What type of outcome?

How many predictor variables?

What type of predictor?

If a categorical predictor, how

many categories?

If a categorical predictor, are the same or different

entities in each category?

Assumptions of linear model met

Continuous

One

Two or more

One

Two or more

One

Two or more

Continuous

Categorical

Continuous

Categorical

Continuous

Categorical

Both

Continuous

Categorical

Both

Categorical

Categorical

Both

Two

More than two

Different

Same

Different

Same

Different

Same

both

Different

Different

Different

Independent t-test or Point-biserial correlation

Bootstrapped t-test or Mann-Whitney Test

Paired-samples t-test (Dependent t-test)

Bootstrapped t-test or Wilcoxon signed-rank test

One-way independent ANOVA

Robust ANOVA or Kruskal-Wallis test

One-way repeated measures ANOVA

Bootstrapped ANOVA or Friedman's ANOVA

Pearson correlation or regression

Bootstrap correlation/regression, Spearman

correlation, Kendall's tau

Factorial repeated measures ANOVA

Robust factorial repeated measures ANOVA

Multiple regression Bootstrapped multiple regression

Factorial mixed ANOVA Robust factorial mixed ANOVA

Independent factorial ANOVA/multiple regression

Robust independent factorial ANOVA/multiple regression

Multiple regression/ANCOVA Robust ANCOVA/bootstrapped regression

Logistic regression or biserial/point biserial

correlation

Pearson chi-square or likelihood ratio

Logistic regression

Loglinear analysis

MANOVA

Factorial MANOVA

MANCOVA

Assumptions of linear model not met

Logistic regression

@ProfAndyField

The GLM

!   The Viagra example from DSUS.

!  We can conduct a study measure a person’s libido and partner’s libido over a period following a dose of Viagra. !   Outcome (or DV) = Participant’s libido !   Predictor (or IV) = Dose of Viagra (continuous

or categorical) !   Covariate = Partner’s libido (continuous)

@ProfAndyField

The Only Equation You’ll Ever Need

Outcome(s)i =Modeli +εi

Models Outcomei = b+εiMean

Outcomei = bPredictori +εiCorrelation

Outcomei = b0 + b1Predictori +εiRegression

Outcomei = b0 + b1Predictori +εit-test

Outcomei = b0 + b1Predictor1i + b2Predictor2i +εiRegression

Outcomei = b0 + b1Predictor1i + b2Predictor2i +εiANOVA

Outcomei = b0 + b1Predictori + b2Predictori +εiANCOVA

Models Libidoi = b+εiMean

Libidoi = bDosei +εiCorrelation

Libidoi = b0 + b1Dosei +εiRegression

Libidoi = b0 + b1Dosei +εit-test

Libidoi = b0 + b1Dosei + b2Partneri +εiRegression

Libidoi = b0 + b1Dose1i + b2Dose2i +εiANOVA

Libidoi = b0 + b1Dosei + b2Partneri +εiANCOVA

Complex Models

Outcomei = b0 + b1Ai + b2Bi + b3ABi +εiModeration

Outcomeij = b0 + b1Predictorij +εij + (u0 j )+ (u1 j )Repeated Measures

Outcomei = b0 + b1Ai + b2Bi + b3ABi +εiFactorial ANOVA

Outcomeij = b0 + b1Predictorij +εij + (u0 j )+ (u1 j )Multilevel

@ProfAndyField

F = t2

@ProfAndyField

Using PENIS in the classroom…

!   The GLM as a framework emphasizes simplicity: !   if you understand one model, you understand them all.

!   You can understand every model by understanding just 5 concepts (PENIS): ! Paramaters

!   Estimation

!   NHST

!   Intervals (of the confidence variety)

!   Standard error

•  Collect some data: •  1, 3, 4, 3, 2 •  Add them up:

P: Parameters

•  Divide by the number of scores, n:

!! = 1+ 3+ 4+ 3+ 2 = 13!

!!!!

!

! = !!!!!!! = 13

5 = 2.6!

@ProfAndyField

The mean as a model

( ) iii errorModelOutcome +=

( )Bellend Dr.

Bellend Dr.Bellend Dr.

error6.21errorFriends

+=

+= X

@ProfAndyField

A Perfect Fit

1

2

3

4

5 ● ● ● ● ● ● ●

1 2 3 4 5 6 7Rater

Rat

ing

(out

of 5

)

@ProfAndyField

1

2

3

4

5 ●● ● ●●●● ●● ● ● ●● ●● ●● ●●● ●● ●●● ●● ●● ●● ●●●● ●●

● ●●● ●● ●●● ● ●●

● ●

●

● ● ●● ● ●● ●●● ●●● ● ●● ● ●● ●

0 20 40 60Customer

Rat

ing

on a

maz

on.c

o.uk

(out

of 5

)

@ProfAndyField

Calculating ‘Error’ !   A deviation is the difference between the

mean and an actual data point.

!   Deviations can be calculated by taking each score and subtracting the mean from it:

xxi −= Deviation

@ProfAndyField

-1.6

-0.6

+0.4 +0.4

+1.4

@ProfAndyField

E: Estimation •  Friends = b*1 + error

•  error = Friends - b

y b Error Squared Error

1 2 -1 1

2 2 0 0

3 2 1 1

3 2 1 1

4 2 2 4

Total 7

Value of b

Sum

of S

quar

ed E

rror

0

5

10

15

20

25

30

35

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

● ● ● ● ● ● ●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

2.6

0 1 2 3 4 5

Population µ = 3

!!= 3

!!= 3

!!= 3

!!= 4

!!= 4

!!= 2

!!= 2

!!= 5

Mean = 3 SD = 1.22

S: Standard Error

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Sperm Count (Millions)

Sample 50Sample 49Sample 48Sample 47Sample 46Sample 45Sample 44Sample 43Sample 42Sample 41Sample 40Sample 39Sample 38Sample 37Sample 36Sample 35Sample 34Sample 33Sample 32Sample 31Sample 30Sample 29Sample 28Sample 27Sample 26Sample 25Sample 24Sample 23Sample 22Sample 21Sample 20Sample 19Sample 18Sample 17Sample 16Sample 15Sample 14Sample 13Sample 12Sample 11Sample 10Sample 9Sample 8Sample 7Sample 6Sample 5Sample 4Sample 3Sample 2Sample 1

These intervals don't contain

the 'true' value

I: Intervals

Sample Number

Sper

m C

ount

(Milli

ons)

0123456789

101112131415161718192021222324252627282930

0123456789

101112131415161718192021222324252627282930

Probability ... .01

●

●

Probability ... .05 (Equal)

●

●

Sample 1 Sample 2

Probability < .01

●

●

Probability ... .05 (Unequal)

●

●

Sample 1 Sample 2

Probability ≈ .01 Probability < .01

Probability ≈ .05Probability ≈ .05

RE Model

−4 −2 0 2 4 6 8

Difference Between Means

Study 10

Study 9

Study 8

Study 7

Study 6

Study 5

Study 4

Study 3

Study 2

Study 1

3.50 [ 1.06 , 5.94 ]

1.65 [ −1.65 , 4.95 ]

3.07 [ −1.15 , 7.30 ]

3.63 [ 1.09 , 6.16 ]

3.31 [ 0.97 , 5.65 ]

2.29 [ −0.52 , 5.10 ]

3.96 [ 1.11 , 6.82 ]

2.64 [ −1.43 , 6.72 ]

2.68 [ −0.63 , 5.99 ]

1.27 [ −2.39 , 4.94 ]

2.97 [ 2.03 , 3.91 ]

p = .476p = .134p = .214p = .012p = .141p = .007p = .006p = .166p = .349

p = .003

@ProfAndyField

N: NHST !   All parameters have an associated distribution.

!   Therefore, for any parameter, we can work out the probability of getting at least the value we have if a null hypothesis of interest is true (e.g., if b = 0, or b1 = b2).

!   Distributions might have different symbols to represent them, but they mean the same thing.

z

Density

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

−4.00 −3.00 −1.96 −1.00 0.00 1.00 1.65 1.96 3.00 4.00

Probability = .025 Probability = .025

Probability = .95

@ProfAndyField

Snowballing Simplicity !   Homoscedasticity/Homogeneity of variance

!   In large samples do not bias bs (CLT applies), but are important for OLS estimation to be optimal.

!   Bias Standard Errors (hence CIs and ps)

!   Normality (of sampling distribution/residuals) !   b are optimal in OLS estimation when residuals normal. !   CIs and ps of bs not biased if the sample is big enough (CLT)

!   Outliers !   Bias Means and Variance/SS

!   Bias tests of parameters (low power) !   Bias parameter/Effect size estimates

!   Remedies !   Any parameter/CI can be bootstrapped (no need for nonparametric

stats) !   Adjust the data (transform, windzorize, trim)

@ProfAndyField

Summary !   The GLM is a useful framework for teaching statistics

!   It enables you to get your PENIS out in lectures without getting arrested.

!   If you understand the PENIS, you understand any statistical model: !   All models have parameters that define the model and tell

you about your hypothesis !   These parameters are estimated !   They vary in samples !   They tell us about the ‘true’ size of the effect (Cis) !   They can be significance tested !   They (and their CIs and p-values) can be biased !   They (and their CIs) can be bootstrapped

@ProfAndyField

My statistics cult:

@ProfAndyField

www.facebook.com/ProfAndyField

discoveringstatistics.blogspot.co.uk

www.youtube.com/user/profandyfield