Useful Statistical Tools

February 19, 2010

Today’s Class

• Aphorisms• Useful Statistical Tools• Probing Question• Assignments• Surveys

Aphorisms“Get close enough to know the task, but stay far enough to see the patterns.”

"Humor happens, embrace it.“

"Much like improv, prom night, and getting into fights, the key to good contextual inquiry is to always say yes.“

"Learn as though you would never be able to master it; hold it as though you would be in fear of losing it.“

"Until you learn to interpret openly, you open yourself to mis-interpretation.“

"To know an answer, you must ask a question. To know a truth, you must contextually inquire the right question.”

"Your participant does all the hard stuff. All you have to do is talk about it and check your work“

"You cannot learn if you already know, unless you first learn how to forget!“

"Listen to the people around you, including to those you know well -- but listen deeper.“

"Do, or do not. There is no try."

Any guesses

• From those who did not email in?

• Juelaila has won the first cookie• There is one cookie remaining

Aphorisms“Get close enough to know the task, but stay far enough to see the patterns.”

"Humor happens, embrace it.“

"Much like improv, prom night, and getting into fights, the key to good contextual inquiry is to always say yes.“

"Learn as though you would never be able to master it; hold it as though you would be in fear of losing it.“

"Until you learn to interpret openly, you open yourself to mis-interpretation.“

"To know an answer, you must ask a question. To know a truth, you must contextually inquire the right question.”

"Your participant does all the hard stuff. All you have to do is talk about it and check your work“

"You cannot learn if you already know, unless you first learn how to forget!“

"Listen to the people around you, including to those you know well -- but listen deeper.“

"Do, or do not. There is no try."

Aphorisms“Get close enough to know the task, but stay far enough to see the patterns.”

"Humor happens, embrace it.“

"Much like improv, prom night, and getting into fights, the key to good contextual inquiry is to always say yes.“

"Learn as though you would never be able to master it; hold it as though you would be in fear of losing it.“

"Until you learn to interpret openly, you open yourself to mis-interpretation.“

"To know an answer, you must ask a question. To know a truth, you must contextually inquire the right question.”

"Your participant does all the hard stuff. All you have to do is talk about it and check your work“

"You cannot learn if you already know, unless you first learn how to forget!“

"Listen to the people around you, including to those you know well -- but listen deeper.“

"Do, or do not. There is no try."

Cookies!

• "Do, or do not. There is no try.“– Juelaila answered first

• "Until you learn to interpret openly, you open yourself to mis-interpretation.“– No answers

Let’s discuss

• A few of these aphorisms

• Do you think that they help us understand the idea and practice of contextual inquiry better?

Your thoughts?

• “Get close enough to know the task, but stay far enough to see the patterns.”

Your thoughts?

• "Much like improv, prom night, and getting into fights, the key to good contextual inquiry is to always say yes.“

Your thoughts?

• "Until you learn to interpret openly, you open yourself to mis-interpretation.“

Your thoughts?

• "Your participant does all the hard stuff. All you have to do is talk about it and check your work“

Your thoughts?

• "You cannot learn if you already know, unless you first learn how to forget!“

Comments? Questions?

Today’s Class

• Aphorisms• Useful Statistical Tools• Probing Question• Assignments• Surveys

Useful Statistical Tools

• Power Analysis• Meta-Analysis• Imputation

Power Analysis

• A set of methods for determining

• The probability that you will obtain a statistically significant result, assuming a true effect size and sample size of a certain magnitude

Or

• The reverse

• Given a certain true effect size, and a desired probability of obtaining a statistically significant result, what sample size is needed?

Why? When?

• Why might a researcher want to do each type of power analysis?

• When might a researcher want to do each type of power analysis?

When used

• Effect size + Power --> Sample Size– Usually used before running study to pick sample

size

• Effect size + Sample Size --> Power– Usually used after running study to explain to

thesis committee why more subjects are needed

Power analysis

• Can be computed from– “Effect Size”/ Cohen’s d• (M1 – M2)/ (pooled SD, e.g. s)

– r– Difference in two r values– And several other metrics

Power analysis

• Can be computed for– Single-group t-test– Two-group t-test– Paired t-test– F test– Sign test– Etc., etc., etc.

Mathematical Details

• Differ for different statistical tests and metrics• Possible to do this in online power calculators

Sign Test Example(Courtesy of John McDonald)

What is a good value for power?

• Conventionally, power = 0.80 is treated as “good”

• Kind of a magic number

Comments? Questions?

I need 3 volunteers

Play with calculator

• http://www.cs.uiowa.edu/~rlenth/Power/

• Two-sample t-test

Volunteer #1

• If the true effect size is 0.5 ,s how big a sample do you need to achieve Power = 0.8?

Volunteer #2

• If the true effect size is 0.2 ,s how big a sample do you need to achieve Power = 0.8?

Volunteer #3

• If your control condition gains 20 points pre-post• And your experimental condition gains 40 points pre-

post• And the pooled standard deviation is 30 points• And you have 20 students in each condition

• What’s your statistical power?

Comments? Questions?

How can statistical power be increased?

• Both in theory, and in real life

How can statistical power be increased?

• Increase sample size

How can statistical power be increased?

• Increase difference in means– Make your intervention better

How can statistical power be increased?

• Increase difference in means– Make your control condition worse• Some researchers make the mistake of picking a control

condition that’s impossibly good– ScienceAssistments versus

ScienceAssistments, with one less potential IV

• This doesn’t mean you should fish for a control condition that is absurdly awful– DrScheme versus

Learning programming through interpretive dance– Miley’s World versus

Learning math through reading textbooks

How can statistical power be increased?

• Increase difference in means– Make your control condition worse• Some researchers make the mistake of picking a control

condition that’s impossibly good– ScienceAssistments versus

ScienceAssistments, with one less potential IV

• This doesn’t mean you should fish for a control condition that is absurdly awful– DrScheme versus

Learning programming through interpretive dance– Miley’s World versus

Learning math through reading textbooks written in Danish

How can statistical power be increased?

• Reduce standard deviation– What methods have we discussed in class that

could help us do this?

How can statistical power be increased?

• Reduce standard deviation– What methods have we discussed in class that

could help us do this?• Stratification

Comments? Questions?

Meta-Analysis

Meta-Analysis

• Very important point, right up front

• There is meta-analysis

• And then there are the statistical techniques used in meta-analysis– Much broader in application than just classical

meta-analysis!

Meta-Analysis

• In the classic sense, integrating across a set of previous studies, to attempt to find an overall effect size or significance of finding across all those studies

Examples

• Kulik & Kulik (1991) computer-aided instruction does 0.3 s better than traditional instruction

• Cohen, Kulik, & Kulik (1982) found that expert tutors do 2.3 s better than traditional instruction; novice tutors only do 0.4 s better than traditional instruction

Process of doing a meta-analysis

• Find all the studies on topic of interest• Find measure of interest (effect size or

statistical significance)• Integrate across studies

Challenges

• What might make it difficult to• Find all the studies on topic of interest• ?

Challenges to Finding all Studies

• Knowing what terminology to use in literature review – many phenomena have many names– Off-task behavior, Time-on-task, Percent On-Task,

Attention– Gaming the system, Systematic Guessing, Hint

Abuse, Help Abuse, Executive Help-Seeking, Letaxmaning, Off-Task Gaming Behavior, Player Transformation, Goal Structure Misalignment

Challenges to Finding all Studies

• “File-Drawer Effect”– Papers with null results get rejected by conference

program committees and journal reviewers– Papers with null results don’t get submitted in the

first place

Find measure of interest

• Statistical significance– If you can find a p, you can turn it into a Z, and

you’re good to go• Using Z formula in Excel, or a Z-p table

– Set direction on Z to be consistent• E.g. all studies with finding X are positive• All studies with finding not-X are negative

Find measure of interest

• Effect size– Transform values into correlations or Cohen’s d

values

Why might you…

• Why might you want to do meta-analysis on effect size versus statistical significance?

Integrating Across Studies

• Two cases

• Studies are independent

• Studies are non-independent

Studies are Independent

• By far the statistically easier case

Aggregating significance tests

• Stouffer’s Z• For N studies, each with Z value

SZsqrt(N)

Volunteer?

Example

• Five studies on the effects of taking gym class on mathematics performance– Two studies found positive effect of taking gym

class, p= 0.02, p=0.06– Three studies found negative effect of taking gym

class, p=0.05, p=0.11, p=0.75

– One-tailed Z table on the next slide

Z table

Aggregating correlations

• Convert r to Fisher z’

• For N studies, each with z’ value

Sz’ N

• Then convert the result back to r

Why Fisher z’?

• Equal differences between any two Fisher z’ values are equal in significance

• Whereas r is uneven– From r=0.8 to 0.9 is a bigger difference in

significance than r=0.2 to r=0.3– So transformation is necessary to weight all

differences in correlation equally

Volunteer?

Example

• Five studies on the effects of learning computer programming on popularity– Two studies found positive correlation, r = 0.1, r=

0.3– Three studies found negative correlation, r = - 0.8,

r = - 0.6, r = - 0.7

– Fisher z’ table on the next slide

r z' r z' r z' r z' r z' r z'

0.00 0.0000 0.18 0.1820 0.35 0.3654 0.53 0.5901 0.69 0.84800.86 1.2933

0.01 0.0100 0.19 0.1923 0.36 0.3769 0.54 0.6042 0.70 0.86730.87 1.3331

0.02 0.0200 0.20 0.2027 0.37 0.3884 0.55 0.6184 0.71 0.88720.88 1.3758

0.03 0.0300 0.21 0.2132 0.38 0.4001 0.56 0.6328 0.72 0.90760.89 1.4219

0.04 0.0400 0.22 0.2237 0.39 0.4118 0.57 0.6475 0.73 0.92870.90 1.4722

0.05 0.0500 0.23 0.2342 0.40 0.4236 0.58 0.6625 0.74 0.95050.91 1.5275

0.06 0.0601 0.24 0.2448 0.41 0.4356 0.59 0.6777 0.75 0.97300.92 1.5890

0.07 0.0701 0.25 0.2554 0.42 0.4477 0.60 0.6931 0.76 0.99620.93 1.6584

0.08 0.0802 0.26 0.2661 0.43 0.4599 0.61 0.7089 0.77 1.02030.94 1.7380

0.09 0.0902 0.27 0.2769 0.44 0.4722 0.62 0.7250 0.78 1.04540.95 1.8318

0.10 0.1003 0.28 0.2877 0.45 0.4847 0.63 0.7414 0.79 1.07140.96 1.9459

0.11 0.1104 0.29 0.2986 0.46 0.4973 0.64 0.7582 0.80 1.09860.97 2.0923

0.12 0.1206 0.30 0.3095 0.47 0.5101 0.65 0.7753 0.81 1.12700.98 2.2976

0.13 0.1307 0.31 0.3205 0.48 0.5230 0.66 0.7928 0.82 1.15680.99 2.6467

0.14 0.1409 0.32 0.3316 0.49 0.5361 0.67 0.8107 0.83 1.1881

0.15 0.1511 0.33 0.3428 0.50 0.5493 0.68 0.8291 0.84 1.2212

0.16 0.1614 0.34 0.3541 0.51 0.5627 0.85 1.2562

0.17 0.1717 0.52 0.5763

Comments? Questions?

Studies are non-independent

• Generally taken to mean that same sample (at least in part) is involved

• The case where there is non-independence due to similar (or the same) learning materials is generally not considered, as inter-correlation is low and difficult to compute

Math is “complex”

• Strube’s (1985) Adjusted Z is used instead of Stouffer’s Z in these cases– Accounts for correlation of different data points

for the same subject

• Similar approach for effect size

Comments? Questions?

Other Uses of These Techniques

Non-independence in modeling

• Take the case where you are studying whether an EDM model is statistically significantly different than chance– N actions involving M students

• It is extremely invalid to do a statistical significance test involving N actions– Assumes each action is independent of each other action

• But it biases towards non-significance to collapse the N data points into one data point per student

Solution

• Do separate statistical significance test within each student (actions can be treated as independent of each other, once student is accounted for)

• Then use Stouffer’s Z to aggregate across students

To see examples…

• There is not time to discuss the math in detail today, but see examples in– Baker, Corbett, & Aleven (2008)– Baker, Corbett, Roll, & Koedinger (2008)

Comments? Questions?

Imputation

Imputation

• In data sets with large amounts of data per data point– For instance, extremely long surveys or

demographic data• It is common to have small amounts of missing

data in each data point– E.g. variable 17 missing for students 1, 14, 90, 112,

202, 477

In these cases…

• It may be undesirable to throw out every data point that has a missing response– You might end up losing 30-40% of your data, or

more, and biasing your data

• For instance, people who occasionally fail to respond to survey items probably differ systematically from people who dilligently and carefully answer every question

Imputation

• For each data point missing a value• Find a set of “similar” data point that is not

missing that value– Similar data point has low absolute difference

across non-missing variables• Randomly choose one of the non-missing

values to fill in the missing data

Multiple Imputation

• Create 3-10 data sets in this fashion• Then for all the missing data, find the mean

(and SD) across all imputed data sets

• Use the no-longer-missing data in future analyses

An alternative: regression imputation

• Find set of linear regression functions predicting each variable from all other variables

• Use this function to fill in missing data

Advantages? Disadvantages?

• Multiple Imputation• Regression Imputation• Throwing out all data points with missing

variables

Comments? Questions?

Today’s Class

• Aphorisms• Useful Statistical Tools• Probing Question• Assignments• Surveys

Probing Question

• Observation: Relatively few researchers use power analysis when designing their studies.

• Why?• Are they making a mistake?

Today’s Class

• Aphorisms• Useful Statistical Tools• Probing Question• Assignments• Surveys

Assignment #5

• Any questions?

Today’s Class

• Aphorisms• Useful Statistical Tools• Probing Question• Assignments• Surveys