Basic Experimental Design Larry V. Hedges Northwestern University Prepared for the IES Summer Research Training Institute July 8, 2008

Basic Experimental Design

Larry V. HedgesNorthwestern University

Prepared for the IES Summer Research Training Institute July 8, 2008

What is Experimental Design?

Experimental design includes both• Strategies for organizing data collection• Data analysis procedures matched to those data

collection strategies

Classical treatments of design stress analysis procedures based on the analysis of variance (ANOVA)

Other analysis procedure such as those based on hierarchical linear models or analysis of aggregates (e.g., class or school means) are also appropriate

Why Do We Need Experimental Design?

Because of variability

We wouldn’t need a science of experimental design if

• If all units (students, teachers, & schools) were identical

and

• If all units responded identically to treatments

We need experimental design to control variability so that treatment effects can be identified

A Little History

The idea of controlling variability through design has a long history

In 1747 Sir James Lind’s studies of scurvy

Their cases were as similar as I could have them. They all in general had putrid gums, spots and lassitude, with weakness of their knees. They lay together on one place … and had one diet common to all (Lind, 1753, p. 149)

Lind then assigned six different treatments to groups of patients

A Little History

The idea of random assignment was not obvious and took time to catch on

In 1648 von Helmont carried out one randomization in a trial of bloodletting for fevers

In 1904 Karl Pearson suggested matching and alternation in typhoid trials

Amberson, et al. (1931) carried out a trial with one randomization

In 1937 Sir Bradford Hill advocated alternation of patients in trials rather than randomization

Diehl, et al. (1938) carried out a trial that is sometimes referred to as randomized, but it actually used alternation

A Little History

The first modern randomized clinical trial in medicine is usually considered to be the trial of streptomycin for treating tuberculosis

It was conducted by the British Medical Research Council in 1946 and reported in 1948

A Little History

Experiments have been used longer in the behavioral sciences (e.g., psychophysics: Pierce and Jastrow, 1885)

Experiments conducted in laboratory settings were widely used in educational psychology (e.g., McCall, 1923)

Thorndike (early 1900’s)

Lindquist (1953)

Gage field experiments on teaching (1978 – 1984)

A Little History

Studies in crop variation I – VI (1921 – 1929)

In 1919 a statistician named Fisher was hired at Rothamsted agricultural station

They had a lot of observational data on crop yields and hoped a statistician could analyze it to find effects of various treatments

All he had to do was sort out the effects of confounding variables

Studies in Crop Variation I (1921)

Fisher does regression analyses—lots of them—to study (and get rid of) the effects of confounders

• soil fertility gradients• drainage• effects of rainfall• effects of temperature and weather, etc.

Fisher does qualitative work to sort out anomalies

Conclusion

The effects of confounders are typically larger than those of the systematic effects we want to study

Studies in Crop Variation II (1923)

Fisher invents

• Basic principles of experimental design

• Control of variation by randomization

• Analysis of variance

Studies in Crop Variation IV and VI

Studies in Crop variation IV (1927)

Fisher invents analysis of covariance to combine statistical control and control by randomization

Studies in crop variation VI (1929)

Fisher refines the theory of experimental design, introducing most other key concepts known today

Our Hero in 1929

Principles of Experimental Design

Experimental design controls background variability so that systematic effects of treatments can be observed

Three basic principles

1. Control by matching

2. Control by randomization

3. Control by statistical adjustment

Their importance is in that order

Control by Matching

Known sources of variation may be eliminated by matching

Eliminating genetic variationCompare animals from the same litter of mice

Eliminating district or school effectsCompare students within districts or schools

However matching is limited• matching is only possible on observable characteristics• perfect matching is not always possible• matching inherently limits generalizability by removing (possibly

desired) variation

Control by Matching

Matching ensures that groups compared are alike on specific known and observable characteristics (in principle, everything we have thought of)

Wouldn’t it be great if there were a method of making groups alike on not only everything we have thought of, but everything we didn’t think of too?

There is such a method

Control by Randomization

Matching controls for the effects of variation due to specific observable characteristics

Randomization controls for the effects all (observable or non-observable, known or unknown) characteristics

Randomization makes groups equivalent (on average) on all variables (known and unknown, observable or not)

Randomization also gives us a way to assess whether differences after treatment are larger than would be expected due to chance.


Random assignment is not assignment with no particular rule. It is a purposeful process

Assignment is made at random. This does not mean that the experimenter writes down the names of the varieties in any order that occurs to him, but that he carries out a physical experimental process of randomization, using means which shall ensure that each variety will have an equal chance of being tested on any particular plot of ground (Fisher, 1935, p. 51)


Random assignment of schools or classrooms is not assignment with no particular rule. It is a purposeful process

Assignment of schools to treatments is made at random. This does not mean that the experimenter assigns schools to treatments in any order that occurs to her, but that she carries out a physical experimental process of randomization, using means which shall ensure that each treatment will have an equal chance of being tested in any particular school (Hedges, 2007)

Control by Statistical Adjustment

Control by statistical adjustment is a form of pseudo-matching

It uses statistical relations to simulate matching

Statistical control is important for increasing precision but should not be relied upon to control biases that may exist prior to assignment

Statistical control is the weakest of the three experimental design principles because its validity depends on knowing a statistical model for responses

Using Principles of Experimental Design

You have to know a lot (be smart) to use matching and statistical control effectively

You do not have to be smart to use randomization effectively

But

Where all are possible, randomization is not as efficient (requires larger sample sizes for the same power) as matching or statistical control

Basic Ideas of Design:Independent Variables (Factors)

The values of independent variables are called levels

Some independent variables can be manipulated, others can’t

Treatments are independent variables that can be manipulated

Blocks and covariates are independent variables that cannot be manipulated

These concepts are simple, but are often confused

Remember: You can randomly assign treatment levels but not blocks

Basic Ideas of Design (Crossing)

Relations between independent variables

Factors (treatments or blocks) are crossed if every level of one factor occurs with every level of another factor

ExampleThe Tennessee class size experiment assigned students to

one of three class size conditions. All three treatment conditions occurred within each of the participating schools

Thus treatment was crossed with schools

Basic Ideas of Design (Nesting)

Factor B is nested in factor A if every level of factor B occurs within only one level of factor A

ExampleThe Tennessee class size experiment actually assigned

classrooms to one of three class size conditions. Each classroom occurred in only one treatment condition

Thus classrooms were nested within treatments

(But treatment was crossed with schools)

Where Do These Terms Come From?(Nesting)

An agricultural experiment where blocks are literally blocks or plots of land

Here each block is literally nested within a treatment condition

Blocks

1 2 … n

T1 T2 … T1

Where Do These Terms Come From?(Crossing)

An agricultural experiment

Blocks were literally blocks of land and plots of land within blocks were assigned different treatments

Blocks

1 2 … n

T1 T2…

T1

T2 T1 T2


Blocks were literally blocks of land and plots of land within blocks were assigned different treatments.

Here treatment literally crosses the blocks

Blocks

1 2 … n

T1 T2…

T1

T2 T1 T2


The experiment is often depicted like this. What is wrong with this as a field layout?

Consider possible sources of bias

Blocks

1 2 … n

Treatment 1

…

Treatment 2

Think About These Designs

A study assigns a reading treatment (or control) to children in 20 schools. Each child is classified into one of three groups with different risk of reading failure.

A study assigns T or C to 20 teachers. The teachers are in five schools, and each teacher teaches 4 science classes

Two schools in each district are picked to participate. Each school has two grade 4 teachers. One of them is assigned to T, the other to C.

Three Basic Designs

The completely randomized designTreatments are assigned to individuals

The randomized block designTreatments are assigned to individuals within blocks(This is sometimes called the matched design, because individuals are matched within blocks)

The hierarchical designTreatments are assigned to blocks, the same treatment is assigned to all individuals in the block

The Completely Randomized Design

Individuals are randomly assigned to one of two treatments

Treatment Control

Individual 1 Individual 1


… …

Individual nT Individual nC

The Randomized Block Design

Block 1 … Block m

Treatment 1

Individual 1

…

…

Individual 1

… …

Individual n1 Individual nm

Treatment 2

Individual n1 +1 Individual nm + 1

… …

Individual 2n1 Individual 2nm

The Hierarchical Design

Treatment Control

Block 1 Block m Block m+1 Block 2m

Individual 1

…


…

Individual 1

Individual 2 Individual 2 Individual 2 Individual 2

… … … …

Individual n1 Individual nm Individual nm+1 Individual n2m

Randomization Procedures

Randomization has to be done as an explicit process devised by the experimenter

• Haphazard is not the same as random

• Unknown assignment is not the same as random

• “Essentially random” is technically meaningless

• Alternation is not random, even if you alternate from a random start

This is why R.A. Fisher was so explicit about randomization processes


R.A. Fisher on how to randomize an experiment with small sample size and 5 treatments

A satisfactory method is to use a pack of cards numbered from 1 to 100, and to arrange them in random order by repeated shuffling. The varieties [treatments] are numbered from 1 to 5, and any card such as the number 33, for example is deemed to correspond to variety [treatment] number 3, because on dividing by 5 this number is found as the remainder. (Fisher, 1935, p.51)


You may want to use a table of random numbers, but be sure to pick an arbitrary start point!

Beware random number generators—they typically depend on seed values, be sure to vary the seed value (if they do not do it automatically)

Otherwise you can reliably generate the same sequence of random numbers every time

It is no different that starting in the same place in a table of random numbers


Completely Randomized Design (2 treatments, 2n individuals)

Make a list of all individuals

For each individual, pick a random number from 1 to 2 (odd or even)

Assign the individual to treatment 1 if even, 2 if odd

When one treatment is assigned n individuals, stop assigning more individuals to that treatment


Completely Randomized Design (2pn individuals, p treatments)

Make a list of all individuals

For each individual, pick a random number from 1 to p

One way to do this is to get a random number of any size, divide by p, the remainder R is between 0 and (p – 1), so add 1 to the remainder to get R + 1

Assign the individual to treatment R + 1

Stop assigning individuals to any treatment after it gets n individuals


Randomized Block Design with 2 Treatments(m blocks per treatment, 2n individuals per block)

Make a list of all individuals in the first block

For each individual, pick a random number from 1 to 2 (odd or even)

Assign the individual to treatment 1 if even, 2 if odd

Stop assigning a treatment it is assigned n individuals in the block

Repeat the same process with every block


Randomized Block Design with p Treatments(m blocks per treatment, pn individuals per block)

Make a list of all individuals in the first block

For each individual, pick a random number from 1 to p

Assign the individual to treatment p

Stop assigning a treatment it is assigned n individuals in the block

Repeat the same process with every block


Hierarchical Design with 2 Treatments(m blocks per treatment, n individuals per block)

Make a list of all blocks

For each block, pick a random number from 1 to 2

Assign the block to treatment 1 if even, treatment 2 if odd

Stop assigning a treatment after it is assigned m blocks

Every individual in a block is assigned to the same treatment


Hierarchical Design with p Treatments(m blocks per treatment, n individuals per block)

Make a list of all blocks

For each block, pick a random number from 1 to p

Assign the block to treatment corresponding to the number

Stop assigning a treatment after it is assigned m blocks

Every individual in a block is assigned to the same treatment

Sampling Models

Sampling Models in Educational Research

Sampling models are often ignored in educational research

But

Sampling is where the randomness comes from in social research

Sampling therefore has profound consequences for statistical analysis and research designs


Simple random samples are rare in field research

Educational populations are hierarchically nested:

• Students in classrooms in schools

• Schools in districts in states

We usually exploit the population structure to sample students by first sampling schools

Even then, most samples are not probability samples, but they are intended to be representative (of some population)


Survey research calls this strategy multistage (multilevel) clustered sampling

We often sample clusters (schools) first then individuals within clusters (students within schools)

This is a two-stage (two-level) cluster sample

We might sample schools, then classrooms, then students

This is a three-stage (three-level) cluster sample

Precision of Estimates Depends on the Sampling Model

Suppose the total population variance is σT2 and ICC is ρ

Consider two samples of size N = mn

A simple random sample or stratified sample

The variance of the mean is σT2/mn

A clustered sample of n students from each of m schools

The variance of the mean is (σT2/mn)[1 + (n – 1)ρ]

The inflation factor [1 + (n – 1)ρ] is called the design effect


Suppose the population variance is σT2

School level ICC is ρS, class level ICC is ρC

Consider two samples of size N = mpn

A simple random sample or stratified sample

The variance of the mean is σT2/mpn

A clustered sample of n students from p classes in m schools

The variance is (σT2/mpn)[1 + (pn – 1)ρS + (n – 1)ρC]

The three level design effect is [1 + (pn – 1)ρS + (n – 1)ρC]


Treatment effects in experiments and quasi-experiments are mean differences

Therefore precision of treatment effects and statistical power will depend on the sampling model


The fact that the population is structured does not mean the sample is must be a clustered sample

Whether it is a clustered sample depends on:

• How the sample is drawn (e.g., are schools sampled first then individuals randomly within schools)

• What the inferential population is (e.g., is the inference to these schools studied or a larger population of schools)


A necessary condition for a clustered sample is that it is drawn in stages using population subdivisions

• schools then students within schools

• schools then classrooms then students

However, if all subdivisions in a population are present in the sample, the sample is not clustered, but stratified

Stratification has different implications than clustering

Whether there is stratification or clustering depends on the definition of the population to which we draw inferences (the inferential population)


The clustered/stratified distinction matters because it influences the precision of statistics estimated from the sample

If all population subdivisions are included in the every sample, there is no sampling (or exhaustive sampling) of subdivisions

• therefore differences between subdivisions add no uncertainty to estimates

If only some population subdivisions are included in the sample, it matters which ones you happen to sample

• thus differences between subdivisions add to uncertainty

Inferential Population and Inference Models

The inferential population or inference model has implications for analysis and therefore for the design of experiments

Do we make inferences to the schools in this sample or to a larger population of schools?

Inferences to the schools or classes in the sample are called conditional inferences

Inferences to a larger population of schools or classes are called unconditional inferences

Inferential Population and Inference Models

Note that the inferences (what we are estimating) are different in conditional versus unconditional inference models

• In a conditional inference, we are estimating the mean (or treatment effect) in the observed schools

• In unconditional inference we are estimating the mean (or treatment effect) in the population of schools from which the observed schools are sampled

We are still estimating a mean (or a treatment effect) but they are different parameters with different uncertainties

Fixed and Random Effects

When the levels of a factor (e.g., particular blocks included) in a study are sampled and the inference model is unconditional, that factor is called random and its effects are called random effects

When the levels of a factor (e.g., particular blocks included) in a study constitute the entire inference population and the inference model is conditional, that factor is called fixed and its effects are called fixed effects

Applications to Experimental Design

We will look in detail at the two most widely used experimental designs in education

• Randomized blocks designs

• Hierarchical designs

Experimental Designs

For each design we will look at

• Structural Model for data (and what it means)

• Two inference models– What does ‘treatment effect’ mean in principle– What is the estimate of treatment effect– How do we deal with context effects

• Two statistical analysis procedures– How do we estimate and test treatment effects– How do we estimate and test context effects– What is the sensitivity of the tests


The population (the sampling frame)

We wish to compare two treatments

• We assign treatments within schools

• Many schools with 2n students in each

• Assign n students to each treatment in each school


The experiment

Compare two treatments in an experiment

• We assign treatments within schools

• With m schools with 2n students in each

• Assign n students to each treatment in each school


Diagram of the design

Schools

Treatment 1 2 … m

1 …

2 …


School 1

Schools

Treatment 1 2 … m

1 …

2 …

The Conceptual Model

The statistical model for the observation on the kth person in the jth school in the ith treatment is

Yijk = μ +αi + βj + αβij + εijk

where

μ is the grand mean,

αi is the average effect of being in treatment i,

βj is the average effect of being in school j,

αβij is the difference between the average effect of treatment i and the effect of that treatment in school j,

εijk is a residual

Effect of Context

ijk i j ij ijkY

Context Effect

Two-level Randomized Block DesignWith No Covariates (HLM Notation)

Level 1 (individual level)

Yijk = β0j + β1jTijk+ εijk ε ~ N(0, σW2)

Level 2 (school Level)

β0j = π00 + ξ0j ξ0j ~ N(0, σS2)

β1j = π10+ ξ1j ξ1j ~ N(0, σTxS2)

If we code the treatment Tijk = ½ or - ½ , then the parameters are identical to those in standard ANOVA

Effects and EstimatesThe population mean of treatment 1 in school j is

α1 + αβ1j

The population mean of treatment 2 in school j is

α2 + αβ2j

The estimate of the mean of treatment 1 in school j is

α1 + αβ1j + ε1j●

The estimate of the mean of treatment 2 in school j is

α2 + αβ2j + ε2j●

Effects and EstimatesThe comparative treatment effect in any given school j is

(α1 – α2) + (αβ1j – αβ2j)

The estimate of comparative treatment effect in school j is

(α1 – α2) + (αβ1j – αβ2j) + (ε1j● – ε2j●)

The mean treatment effect in the experiment is

(α1 – α2) + (αβ1● – αβ2●)

The estimate of the mean treatment effect in the experiment is

(α1 – α2) + (αβ 1● – αβ2●) + (ε1●● – ε2●●)

Inference Models

Two different kinds of inferences about effects

Unconditional Inference (Schools Random)Inference to the whole universe of schools(requires a representative sample of schools)

Conditional Inference (Schools Fixed)Inference to the schools in the experiment(no sampling requirement on schools)

Statistical Analysis Procedures

Two kinds of statistical analysis procedures

Mixed Effects Procedures (Schools Random)Treat schools in the experiment as a sample from a population of schools(only strictly correct if schools are a sample)

Fixed Effects Procedures (Schools Fixed)Treat schools in the experiment as a population

Unconditional Inference(Schools Random)


(α1 – α2) + (αβ 1● – αβ2●) + (ε1●● – ε2●●)

The average treatment effect we want to estimate is

(α1 – α2)

The term (ε1●● – ε2●●) depends on the students in the schools in the sample

The term (αβ1● – αβ2●) depends on the schools in sample

Both (ε1●● – ε2●●) and (αβ1● – αβ2●) are random and average to 0 across students and schools, respectively

Conditional Inference(Schools Fixed)

The estimate of the mean treatment effect in the experiment is still

(α1 – α2) + (αβ 1● – αβ2●) + (ε1●● – ε2●●)

Now the average treatment effect we want to estimate is

(α1 + αβ1●) – (α2 + αβ2●) = (α1 – α2) + (αβ1● – αβ2●)


The term (αβ1● – αβ2●) depends on the schools in sample, but the treatment effect in the sample of schools is the effect we want to estimate

Expected Mean SquaresRandomized Block Design

(Two Levels, Schools Random)

Source df E{MS}

Treatment (T) 1 σW2 + nσTxS

2 + nmΣαi2

Schools (S) m – 1 σW2 + 2nσS

2

T X S m – 1 σW2 + nσTxS

2

Within Cells 2 m(n – 1) σW2

Mixed Effects Procedures(Schools Random)

The test for treatment effects has

H0: (α1 – α2) = 0

Estimated mean treatment effect in the experiment is

(α1 – α2) + (αβ1● – αβ2●) + (ε1●● – ε2●●)

The variance of the estimated treatment effect is

2[σW2 + nσTxS

2] /mn = 2[1 + (nωS – 1)ρ]σ2/mn

Here ωS = σTxS2/σS

2 and ρ = σS2/(σS

2 + σW2) = σS

2/σ2

Mixed Effects Procedures

The test for treatment effects:

FT = MST/MSTxS with (m – 1) df

The test for context effects (treatment by schools interaction) is

FTxS = MSTxS/MSWS with 2m(n – 1) df

Power is determined by the operational effect size

where ωS = σTxS2/σS

2 and ρ = σS2/(σS

2 + σW2) = σS

2/σ2

1 2

1 ( 1)S

α α n

nω ρ

Expected Mean SquaresRandomized Block Design

(Two Levels, Schools Fixed)

Source df E{MS}

Treatment (T) 1 σW2 + nmΣαi

2

Schools (S) m – 1 σW2 + 2nΣβi

2/(m – 1)

S X T m – 1 σW2 + nΣΣαβij

2/(m – 1)

Within Cells 2m(n – 1) σW2

Fixed Effects Procedures


H0: (α1 – α2) + (αβ1● – αβ2●) = 0


(α1 – α2) + (αβ1● – αβ2●) + (ε1●● – ε2●●)


2σW2 /mn

Fixed Effects ProceduresThe test for treatment effects:

FT = MST/MSWS with m(n – 1) df

The test for context effects (treatment by schools interaction) is

FC = MSTxS/MSWS with 2m(n – 1) df


with m(n – 1) df

1 2 1 2α α α αn

Comparing Fixed and Mixed Effects Statistical Procedures

(Randomized Block Design)

Fixed Mixed

Inference Model Conditional Unconditional

Estimand (α1 – α2) + (αβ1● – αβ2●) (α1 – α2)

Contaminating Factors (ε1●● – ε2●●) (αβ1● – αβ2●) + (ε1●● – ε2●●)

Operational Effect Size

df 2m(n – 1) (m – 1)

Power higher lower

1 2 1 2α α α αn

1 2

1 ( 1)S

α α n

nω ρ

Comparing Fixed and Mixed Effects Procedures(Randomized Block Design)

Conditional and unconditional inference models • estimate different treatment effects• have different contaminating factors that add uncertainty

Mixed procedures are good for unconditional inference

The fixed procedures are good for conditional inference

The fixed procedures have higher power


The universe (the sampling frame)


• We assign treatments to whole schools• Many schools with n students in each• Assign all students in each school to the

same treatment


The experiment


• We assign treatments to whole schools• Assign 2m schools with n students in each• Assign all students in each school to the

same treatment


Diagram of the experiment

Schools

Treatment 1 2 … m m +1 m +2 … 2 m

1

2


Treatment 1 schools

Schools

Treatment 1 2 … m m +1 m + 2 … 2 m

1

2


Treatment 2 schools

Schools

Treatment 1 2 … m m + 1 m + 2 … 2 m

1

2



Yijk = μ + αi + βi + αβij + εjk(i) = μ + αi + βj(i) + εjk(i)



βj is the average effect if being in school j,


εijk is a residual

Or βj(i) = βi + αβij is a term for the combined effect of schools within treatments



Yijk = μ + αi + βi + αβij + εjk(i) = μ + αi + βj(i) + εjk(i)



βj is the average effect if being in school j,


εijk is a residual

or βj(i) = βi + αβij is a term for the combined effect of schools within treatments

Context Effects

Two-level Hierarchical DesignWith No Covariates (HLM Notation)


Yijk = β0j + εijk ε ~ N(0, σW2)


γ0j = π00 + π01Tj + ξ0j ξ ~ N(0, σS2)

If we code the treatment Tj = ½ or - ½ , then

π00 = μ, π01 = α1, ξ0j = βj(i)

The intraclass correlation is ρ = σS2/(σS

2 + σW2) = σS

2/σ2

Effects and EstimatesThe comparative treatment effect in any given school j is still

(α1 – α2) + (αβ1j – αβ2j)

But we cannot estimate the treatment effect in a single school because each school gets only one treatment

The mean treatment effect in the experiment is

(α1 – α2) + (β●(1) – β●(2))

= (α1 – α2) +(β1● – β2● )+ (αβ1● – αβ2●)


(α1 – α2) + (β● (1) – β● (2)) + (ε1●● – ε2●●)

Inference Models

Two different kinds of inferences about effects (as in the randomized block design)

Unconditional Inference (schools random)Inference to the whole universe of schools(requires a representative sample of schools)

Conditional Inference (schools fixed)Inference to the schools in the experiment(no sampling requirement on schools)

Unconditional Inference(Schools Random)

The average treatment effect we want to estimate is

(α1 – α2)


The term (β●(1) – β●(2)) depends on the schools in sample

Both (ε1●● – ε2●●) and (β●(1) – β●(2)) are random and average to 0 across students and schools, respectively

Conditional Inference(Schools Fixed)

The average treatment effect we want to (can) estimate is

(α1 + β●(1)) – (α2 + β●(2)) = (α1 – α2) + (β●(1) – β●(2))

= (α1 – α2) + (β1● – β2● )+ (αβ1● – αβ2●)

The term (β●(1) – β●(2)) depends on the schools in sample, but we want to estimate the effect of treatment in the schools in the sample

Note that this treatment effect is not quite the same as in the randomized block design, where we estimate (α1 – α2) + (αβ1● – αβ2●)

Statistical Analysis Procedures

Two kinds of statistical analysis procedures (as in the randomized block design)

Mixed Effects Procedures

Treat schools in the experiment as a sample from a universe

Fixed Effects Procedures

Treat schools in the experiment as a universe

Expected Mean SquaresHierarchical Design

(Two Levels, Schools Random)

Source df E{MS}

Treatment (T) 1 σW2 + nσS

2 + nmΣαi2

Schools (S) 2(m – 1) σW2 + nσS

2

Within Schools 2m(n – 1) σW2



H0: (α1 – α2) = 0


(α1 – α2) + (β●(1) – β●(2)) + (ε1●● – ε2●●)


2[σW2 + nσS

2] /mn = 2[1 + (n – 1)ρ]σ2/mn

where ρ = σS2/(σS

2 + σW2) = σS

2/σ2



FT = MST/MSBS with (m – 2) df

There is no omnibus test for context effects


where ρ = σS2/(σS

2 + σW2) = σS

2/σ2

1 2

1 ( 1)

α α n

n ρ

Expected Mean SquaresHierarchical Design

(Two Levels, Schools Fixed)

Source df E{MS}

Treatment (T) 1 σW2 + nmΣ(αi + β●(i))2

Schools (S) m – 1 σW2 + nΣΣβj(i)

2/2(m – 1)

Within Schools 2 m(n – 1) σW2

Mixed Effects Procedures(Schools Fixed)


H0: (α1 – α2) + (β●(1) – β●(2)) = 0

Note that the school effects are confounded with treatment effects


(α1 – α2) + (β●(1) – β●(2)) + (ε1●● – ε2●●)

The variance of the estimated treatment effect is 2σW

2 /mn

Mixed Effects Procedures(Schools Fixed)


FT = MST/MSWS with m(n – 1) df

There is no omnibus test for context effects, because each school gets only one treatment


and m(n – 1) df

1 2 (1) (2)α αn

Comparing Fixed and Mixed Effects Procedures(Hierarchical Design)

Fixed Mixed

Inference Model

Conditional Unconditional

Estimand (α1 – α2) + (β●(1) – β●(2)) (α1 – α2)

Contaminating Factors

(ε1●● – ε2●●) (β●(1) – β●(2)) + (ε1●● – ε2●●)

Effect Size

df m(n – 1) (m – 2)

Power higher lower

1 2

1 ( 1)

α α n

n ρ

1 2 (1) (2)α α

n

Comparing Fixed and Mixed Effects Statistical Procedures (Hierarchical Design)

Conditional and unconditional inference models • estimate different treatment effects• have different contaminating factors that add uncertainty

Mixed procedures are good for unconditional inference

The fixed procedures are not generally recommended

The fixed procedures have higher power

Comparing Hierarchical Designs to Randomized Block Designs

Randomized block designs usually have higher power, but assignment of different treatments within schools or classes may be

• practically difficult• politically infeasible• theoretically impossible

It may be methodologically unwise because of potential for

• Contamination or diffusion of treatments• compensatory rivalry or demoralization

Applications to Experimental Design

We will address the two most widely used experimental designs in education

• Randomized blocks designs with 2 levels

• Randomized blocks designs with 3 levels

• Hierarchical designs with 2 levels

• Hierarchical designs with 3 levels

We also examine the effect of covariates

Hereafter, we generally take schools to be random

Complications

Which matchings do we have to take into account in design (e.g., schools, districts, regions, states, regions of the country, country)?

Ignore some, control for effects of others as fixed blocking factors

Justify this as part of the population definition

For example, we define the inference population as these five districts within these two states

But, doing so obviously constrains generalizability

Precision of the Estimated Treatment Effect

Precision is the standard error of the estimated treatment effect

Precision in simple (simple random sample) designs depends on:

• Standard deviation in the population σ

• Total sample size N

The precision is

2SE N

Precision of the Estimated Treatment Effect

Precision in complex (clustered sample) designs depends on:

• The (total) standard deviation σT

• Sample size at each level of sampling (e.g., m clusters, n individuals per cluster)

• Intraclass correlation structure

It is a little harder to compute than in simple designs, but important because it helps you see what matters in design

Intraclass Correlations inTwo-level Designs

In two-level designs the intraclass correlation structure is determined by a single intraclass correlation

This intraclass correlation is the proportion of the total variance that is between schools (clusters)

2 2

2 2 2

S S

S W T

ρ

Precision in Two-level Hierarchical DesignWith No Covariates

The standard error of the treatment effect is

SE decreases as m (number of schools) increases

SE deceases as n increases, but only up to point

SE increases as ρ increases

2 1 ( 1)T

n ρSE

m n

Statistical Power

Power in simple (simple random sample) designs depends on:

• Significance level

• Effect size

• Sample size

Look power up in a table for sample size and effect size

Fragment of Cohen’s Table 2.3.5

d

n 0.10 0.20 … 0.80 1.00 1.20 1.40

8 05 07 … 31 46 60 73

9 06 07 … 35 51 65 79

10 06 07 … 39 56 71 84

11 06 07 … 43 63 76 87

Computing Statistical Power

Power in complex (clustered sample) designs depends on: • Significance level

• Effect size δ

• Sample size at each level of sampling (e.g., m clusters, n individuals per cluster)

• Intraclass correlation structure

This makes it seem a lot harder to compute

Computing Statistical Power

Computing statistical power in complex designs is only a little harder than computing it for simple designs

Compute operational effect size (incorporates sample design information) ΔT

Look power up in a table for operational sample size and operational effect size

This is the same table that you use for simple designs

Power in Two-level Hierarchical DesignWith No Covariates

Basic Idea:Operational Effect Size = (Effect Size) x (Design Effect)

ΔT = δ x (Design Effect)

For the two-level hierarchical design with no covariates

Operational sample size is number of schools (clusters)

1 1

T n

n ρ

1 1

T n

n ρ

Power in Two-level Hierarchical DesignWith No Covariates

As m (number of schools) increases, power increases

As effect size increases, power increases

Other influences occur through the design effect

As ρ increases the design effect (and power) decreases

No matter how large n gets the maximum design effect is

Thus power only increases up to some limit as n increases

1 1

1

1 1 (1 )n n

n

n ρ

1/ ρ

Two-level Hierarchical DesignWith Covariates (HLM Notation)


Yijk = β0j + β1jXijk+ εijk ε ~ N(0, σAW2)


β0j = π00 + π01Tj + π02Wj + ξ0j ξ ~ N(0, σAS2)

β1j = π10

Note that the covariate effect β1j = π10 is a fixed effect

If we code the treatment Tj = ½ or - ½ , then the parameters are identical to those in standard ANCOVA

Precision in Two-level Hierarchical DesignWith Covariates

The standard error of the treatment effect

SE decreases as m increases



SE decreases as RW2 and RS

2 increase

2 21 12

2W S W

T

n ρ R nR RSE

m n

Power in Two-level Hierarchical DesignWith Covariates

Basic Idea:

Operational Effect Size = (Effect Size) x (Design Effect)


For the two-level hierarchical design with covariates

The covariates increase the design effect

2 21 1

TA 2

W S W

n

n ρ R nR R

Power in Two-level Hierarchical DesignWith Covariates

As m and effect size increase, power increases


As ρ increases the design effect (and power) decrease

Now the maximum design effect as large n gets big is

As the covariate-outcome correlations RW2 and RS

2 increase the design effect (and power) increases

21 (1 )SR ρ

2 21 1 2W S W

n

n ρ R nR R

Three-level Hierarchical Design

Here there are three factors• Treatment• Schools (clusters) nested in treatments• Classes (subclusters) nested in schools

Suppose there are• m schools (clusters) per treatment• p classes (subclusters) per school (cluster)• n students (individuals) per class (subcluster)

Three-level Hierarchical DesignWith No Covariates

The statistical model for the observation on the lth person in the kth class in the jth school in the ith treatment is

Yijkl = μ + αi + βj(i) + γk(ij) + εijkl

where μ is the grand mean, αi is the average effect of being in treatment i, βj(i) is the average effect of being in school j, in treatment i γk(ij) is the average effect of being in class k in treatment i, in

school j, εijkl is a residual

Three-level Hierarchical DesignWith No Covariates (HLM Notation)


Yijkl = β0jk + εijkl ε ~ N(0, σW2)

Level 2 (classroom level)

β0jk = γ0j + η0jk η ~ N(0, σC2)


γ0j = π00 + π01Tj + ξ0j ξ ~ N(0, σS2)


π00 = μ, π01 = α1, ξ0j = γk(ij), η0jk = βj(i)

Three-level Hierarchical Design Intraclass Correlations

In three-level designs there are two levels of clustering and two intraclass correlations

At the school (cluster) level

At the classroom (subcluster) level

2 2

2 2 2 2S S

SS C W T

ρ

2 2

2 2 2 2C C

CS C W T

ρ

Precision in Three-level Hierarchical DesignWith No Covariates



SE deceases as p and n increase, but only up to point

SE increases as ρS and ρC increase

1 1 ( 1)2 S CT

pn ρ nSE

m pn

Power in Three-level Hierarchical DesignWith No Covariates

Basic Idea:



For the three-level hierarchical design with no covariates

The operational sample size is the number of schools

1 ( 1) 1

T

S C

pn

pn n ρ

Power in Three-level Hierarchical DesignWith No Covariates

As m and the effect size increase, power increases


As ρS or ρC increases the design effect decreases



1 ( 1) 1 S C

pn

pn n ρ

11 S Cp

Three-level Hierarchical DesignWith Covariates (HLM Notation)


Yijkl = β0jk + β1jkXijkl + εijkl ε ~ N(0, σAW2)


β0jk = γ00j + γ01jZjk + η0jk η ~ N(0, σAC2)

β1jk = γ10j


γ00j = π00 + π01Tj + π02Wj + ξ0j ξ ~ N(0, σAS2)

γ01j = π01

γ10j = π10

The covariate effects β1jk = γ10j = π10 and γ01j = π01 are fixed

Precision in Three-level Hierarchical DesignWith Covariates




SE decreases as RW2, RC

2, and RS2 increase

2 2 2 2

2

1 ( 1) 1

T

2S W S W S C W C

SEm

pn n ρ R pnR R nR R

pn

Power in Three-level Hierarchical DesignWith Covariates

Basic Idea:



For the three-level hierarchical design with covariates


2 2 2 21 ( 1) 1

TA 2

S W S W S C W C

pn


Power in Three-level Hierarchical DesignWith Covariates



As ρS or ρC increase the design effect decreases



2 211 1 1S S C CpR R

2 2 2 21 ( 1) 1 2

S W S W S C W C

pn


Randomized Block Designs

Two-level Randomized Block DesignWith No Covariates (HLM Notation)


Yijk = β0j + β1jTijk+ εijk ε ~ N(0, σW2)


β0j = π00 + ξ0j ξ0j ~ N(0, σS2)

β1j = π10+ ξ1j ξ1j ~ N(0, σTxS2)

If we code the treatment Tijk = ½ or - ½ , then the parameters are identical to those in standard ANOVA

Randomized Block Designs

In randomized block designs, as in hierarchical designs, the intraclass correlation has an impact on precision and power

However, in randomized block designs designs there is also a parameter reflecting the degree of heterogeneity of treatment effects across schools

We define this heterogeneity parameter ωS in terms of the amount of heterogeneity of treatment effects relative to the heterogeneity of school means

Thus

ωS = σTxS2/σS

2

Precision in Two-level Randomized Block DesignWith No Covariates


SE decreases as m (number of schools) increases

SE deceases as n and p increase, but only up to point


SE increases as ωS = σTxS2/σS

2 increases

1 ( 1)2 S

Tn ρ

SEm n

Power in Two-level Randomized Block DesignWith No Covariates

Basic Idea:Operational Effect Size = (Effect Size) x (Design Effect)


For the two-level hierarchical design with no covariates

Operational sample size is number of schools (clusters)

1 1

T n

n ρ

/ 2

1 1

T

S

n

n ρ

Precision in Two-level Randomized Block DesignWith Covariates





SE increases as ωS = σTxS2/σS

2 increases

SE (generally) decreases as RW2 and RS

2 increase

2 21 12

2S W S S W

T

n ρ R n R RSE

m n

Power in Two-level Randomized Block DesignWith Covariates

Basic Idea:



For the two-level hierarchical design with covariates

The covariates increase the design effect

2 2

/ 2

1 1

TA 2

S W S S W

n

n ρ R n R R

Three-level Randomized Block Designs

Three-level Randomized Block DesignWith No Covariates

Here there are three factors• Treatment• Schools (clusters) nested in treatments• Classes (subclusters) nested in schools

Suppose there are• m schools (clusters) per treatment• 2p classes (subclusters) per school (cluster)• n students (individuals) per class (subcluster)

Three-level Randomized Block DesignWith No Covariates

The statistical model for the observation on the lth person in the kth class in the ith treatment in the jth school is

Yijkl = μ +αi + βj + γk + αβij + εijkl

where μ is the grand mean, αi is the average effect of being in treatment i, βj is the average effect of being in school j,

γk is the effect of being in the kth class,αβij is the difference between the average effect of

treatment i and the effect of that treatment in school j, εijkl is a residual

Three-level Randomized Block DesignWith No Covariates (HLM Notation)


Yijkl = β0jk + εijkl ε ~ N(0, σW2)


β0jk = γ00j + γ01jTj + η0jk η ~ N(0, σC2)


γ00j = π00 + ξ0j ξoi ~ N(0, σS2)

γ01j = π10 + ξ1j ξ1i ~ N(0, σTxS2)


π00 = μ, π10 = α1, ξ0j = βj , ξ1j = αβij , η0jk = γk

Three-level Randomized Block Design Intraclass Correlations

In three-level designs there are two levels of clustering and two intraclass correlations

At the school (cluster) level

At the classroom (subcluster) level

2 2

2 2 2 2S S

SS C W T

ρ

2 2

2 2 2 2C C

CS C W T

ρ

Three-level Randomized Block Design Heterogeneity Parameters

In three-level designs, as in two-level randomized block designs, there is also a parameter reflecting the degree of heterogeneity of treatment effects across schools

We define this parameter ωS in terms of the amount of heterogeneity of treatment effects relative to the heterogeneity of school means (just like in two-level designs)

Thus

ωS = σTxS2/σS

2

Precision in Three-level Randomized Block DesignWith No Covariates




SE increases as ωS increases

SE increases as ρS and ρC increase

1 1 ( 1)2 S S CT

pn ρ nSE

m pn

Power in Three-level Randomized Block DesignWith No Covariates

Basic Idea:



For the three-level hierarchical design with no covariates


/ 2

1 ( 1) 1T

S S C

pn

pn n ρ

Power in Three-level Randomized Block DesignWith No Covariates






/ 2

1 ( 1) 1S S C

pn

pn n ρ

11 2 S S Cp

Power in Three-level Randomized Block DesignWith Covariates


SE deceases as p and n increases, but only up to point

SE increases as ρ and ωS increase

SE decreases as RW2, RC

2, and RS2 increase

2 2 2 2

2

1 ( 1) 1

T

2S S W S S W S C W C

SEm

pn n ρ R pn R R nR R

pn


Basic Idea:



For the three-level hierarchical design with covariates


2 2 2 2

/ 2

1 ( 1) 1

TA

2S S W S S W S C W C

pn








2 211 2 1 1S S S C CpR R

2 2 2 2

/ 2

1 ( 1) 1 2

S S W S S W S C W C

pn


What Unit Should Be Randomized?(Schools, Classrooms, or Students)

Experiments cannot estimate the causal effect on any individual

Experiments estimate average causal effects on the units that have been randomized

• If you randomize schools the (average) causal effects are effects on schools

• If you randomize classes, the (average) causal effects are on classes

• If you randomize individuals, the (average) causal effects estimated are on individuals


Theoretical Considerations

Decide what level you care about, then randomize at that level

Randomization at lower levels may impact generalizability of the causal inference (and it is generally a lot more trouble)

Suppose you randomize classrooms, should you also randomly assign students to classes?

It depends: Are you interested in the average causal effect of treatment on naturally occurring classes or on randomly assembled ones?


Relative power/precision of treatment effect

Assign Schools(Hierarchical Design)

Assign Classrooms(Randomized Block)

Assign Students(Randomized Block)

1 1 ( 1)S Cpn ρ n

pn

1 1 ( 1)S S C Cpn ρ n

pn

1 1 ( 1)S S Cpn ρ n

pn


Precision of estimates or statistical power dictate assigning the lowest level possible

But the individual (or even classroom) level will not always be feasible or even theoretically desirable

Questions and Answers About Design


1. Is it ok to match my schools (or classes) before I randomize to decrease variation?

2. I assigned treatments to schools and am not using classes in the analysis. Do I have to take them into account in the design?

3. I am assigning schools, and using every class in the school. Do I have to include classes as a nested factor?

4. My schools all come from two districts, but I am randomly assigning the schools. Do I have to take district into account some way?


1. I didn’t really sample the schools in my experiment (who does?). Do I still have to treat schools as random effects?

2. I didn’t really sample my schools, so what population can I generalize to anyway?

3. I am using a randomized block design with fixed effects. Do you really mean I can’t say anything about effects in schools that are not in the sample?


1. We randomly assigned, but our assignment was corrupted by treatment switchers. What do we do?

2. We randomly assigned, but our assignment was corrupted by attrition. What do we do?

3. We randomly assigned but got a big imbalance on characteristics we care about (gender, race, language, SES). What do we do?

4. We randomly assigned but when we looked at the pretest scores, we see that we got a big imbalance (a “bad randomization”). What do we do?


1. We care about treatment effects, but we really want to know about mechanism. How do we find out if implementation impacts treatment effects?

2. We want to know where (under what conditions) the treatment works. Can we analyze the relation between conditions and treatment effect to find this out?

3. We have a randomized block design and find heterogeneous treatment effects. What can we say about the main effect of treatment in the presence of interactions?


1. I prefer to use regression and I know that regression and ANOVA are equivalent. Why do I need all this ANOVA stuff to design and analyze experiments?

2. Don’t robust standard errors in regression solve all these problems?

3. I have heard of using “school fixed effects” to analyze a randomized block design. Is the a good alternative to ANOVA or HLM?

4. Can I use school fixed effects in a hierarchical design?


1. We want to use covariates to improve precision, but we find that they act somewhat differently in different groups (have different slopes). What do we do?

2. We get somewhat different variances in different groups. Should we use robust standard errors?

3. We get somewhat different answers with different analyses. What do we do?

Thank You !

Documents

Basic Experimental Design Larry V. Hedges Northwestern University Prepared for the IES Summer Research Training Institute July 8, 2008