Experimental Design Terminology An Experimental Unit is the entity on which

measurement or an observation is made. For example, subjects are experimental units in most studies.

Homogeneous Experimental Units: Units that are as uniform as possible on all characteristics that could affect the response.

A Block is a group of homogeneous experimental units.

A Replication is the repetition of an entire experiment or portion of an experiment under two or more sets of conditions.

A Factor is a controllable independent variable that is thought to influence the response.

Experimental Design Terminology

Factors can be fixed or random Fixed -- the factor can take on a discrete

number of values and these are the only values of interest.

Random -- the factor can take on a wide range of values and one wants to generalize from specific values to all possible values.

Each specific value of a factor is called a level.

Experimental Design Terminology A covariate is an independent variable not

manipulated by the experimenter but still affecting the response.

Effect is the change in the average response between two factor levels.

Interaction is the joint factor effects in which the effect of each factor depends on the level of the other factors.

A Design (layout) of the experiment includes the choice of factors and factor-levels, number of replications, blocking, randomization, and the assignment of factor –level combination to experimental units.

Experimental Design Terminology Sum of Squares (SS): Let x1, …, xn be n observations.

The sum of squares of these n observations can be written as x1

2 + x22 +…. xn

2. In notations, ∑xi2. In a

corrected form this sum of squares can be written as Degrees of freedom (df): Number of quantities of the

form – Number of restrictions. For example, in the following SS, we need n quantities of the form . There is one constraint So the df for this SS is n – 1.

Mean Sum of Squares (MSS): The SS divided by it’s df.

n

ii xx

1

2.)(

n

ii xx

1

.0)(xxi

Experimental Design Terminology The analysis of variance (ANOVA) is a technique of

decomposing the total variability of a response variable into: Variability due to the experimental factor(s) and… Variability due to error (i.e., factors that are not

accounted for in the experimental design). The basic purpose of ANOVA is to test the equality of

several means. A fixed effect model includes only fixed factors in the

model. A random effect model includes only random factors

in the model. A mixed effect model includes both fixed and random

factors in the model.

One-way analysis of Variance One factor of k levels or groups. E.g., 3 treatment groups

in a drug study. The main objective is to examine the equality of means of

different groups. Total variation of observations (SST) can be split in two

components: variation between groups (SSG) and variation within groups (SSE).

Variation between groups is due to the difference in different groups. E.g. different treatment groups or different doses of the same treatment.

Variation within groups is the inherent variation among the observations within each group.

Completely randomized design (CRD) is an example of one-way analysis of variance.

One-way analysis of variance

Consider a layout of a study with 16 subjects that intended to compare 4 treatment groups (G1-G4). Each group contains four subjects.

S1 S2 S3 S4G1 Y11 Y12 Y13 Y14G2 Y21 Y22 Y23 Y24G3 Y31 Y32 Y33 Y34G4 Y41 Y42 Y43 Y44

One-way analysis of Variance Model:

Assumptions: Observations yij are independent. eij are normally distributed with mean zero and

constant standard deviation.

error. theis andmean general theis

group, ofeffect theis

group, ofn observatio theis where,

ij

i

ij

ijiij

e

ith

jthithy

ey

One-way analysis of Variance Hypothesis:

Ho: Means of all groups are equal.

Ha: At least one of them is not equal to other.

Analysis of variance (ANOVA) Table for one way classified data

Sources of Variation

Sum of Squares

df Mean Sum of Squares

F-Ratio

Group SSG k-1 MSG=SSG/k-1

F=MSG/MSE

Error SSE n-k MSE=SSE/n-k

Total SST n-1

Multiple comparisons

If the F test is significant in ANOVA table, then we intend to find the pairs of groups are significantly different. Following are the commonly used procedures:

Fisher’s Least Significant Difference (LSD) Tukey’s method Bonferroni’s method Scheffe’s method

One-way ANOVA - Demo MS Excel:

Put response data (hgt) for each groups (grp) in side by side columns (see next slides)

Select Tools/Data Analysis and select Anova: Single Factor from the Analysis Tools list. Click OK.

Select Input Range (for our example a1: c21), mark on Group by columns and again mark labels in first row.

Select output range and then click on ok.

One-way ANOVA MS-Excel Data Layout

grp1 grp2 grp352.50647 47.83167 43.8221143.14426 43.69665 40.4135955.91853 40.73333 57.6574845.68187 46.56424 41.9620754.76593 53.03273 49.0805145.27999 56.41127 55.9779741.95513 43.69635 47.7141943.67319 50.92119 41.7591244.01857 40.36097 46.2185944.54295 51.95264 53.61966

46.1765 57.1638 49.6348440.02826 50.98321 57.6822958.09449 49.23148 49.0847143.25757 48.01014 40.5986642.07507 45.98231 38.9905546.80839 55.32514 54.7428643.80479 47.21837 53.7462457.60508 40.30104 44.8250742.47022 50.56327 45.85581

57.4945 55.55145 41.36863

One-way ANOVA MS-Excel output: height on treatment groups

Anova: Single Factor

SUMMARYGroups Count Sum Average Variance

grp1 20 949.3017 47.46509 36.88689grp2 20 975.5313 48.77656 28.10855grp3 20 954.7549 47.73775 37.50739

ANOVASource of Variation SS df MS F P-value F critBetween Groups 19.15632 2 9.578159 0.280329 0.756571 3.158846Within Groups 1947.554 57 34.16761

Total 1966.71 59

One-way ANOVA - Demo SPSS:

Select Analyze > Compare Means > One –Way ANOVA

Select variables as Dependent List: response (hgt), and Factor: Group (grp) and then make selections as follows-click on Post Hoc and select Multiple comparisons (LSD, Tukey, Bonferroni, or Scheffe), click options and select Homogeneity of variance test, click continue and then Ok.

One-way ANOVA SPSS output: height on treatment groups

ANOVA hgt

Sum of Squares df Mean Square F Sig.

Between Groups 19.156 2 9.578 .280 .757 Within Groups 1947.554 57 34.168 Total 1966.710 59

One-way ANOVA R output: height on treatment groups>grp<- as.factor(grp)

> summary(aov(hgt~grp))

Df Sum Sq Mean Sq F value Pr(>F)

grp 2 19.16 9.58 0.2803 0.7566

Residuals 57 1947.55 34.17

Analysis of variance of factorial experiment (Two or more factors)

Factorial experiment: The effects of the two or more factors including their interactions are investigated simultaneously. For example, consider two factors A and B. Then total variation of the response will be split into variation for A, variation for B, variation for their interaction AB, and variation due to error.

Analysis of variance of factorial experiment (Two or more factors)

Model with two factors (A, B) and their interactions:

Assumptions: The same as in One-way ANOVA.

error theis

B of level andA level ofeffect n interactio theis

Bfactor theof level ofeffect theis

Afactor theof level ofeffect theis

mean general theis

)(

ijk

ij

j

i

ijkijjiijk

e

jthithβ)(

jth

ithα

ey

Analysis of variance of factorial experiment (Two or more factors)

Null Hypotheses: Hoa: Means of all groups of the factor A

are equal. Hob: Means of all groups of the factor B

are equal. Hoab:(αβ)ij = 0, i. e. two factors A and B

are independent

Analysis of variance of factorial experiment (Two or more factors)

ANOVA for two factors A and B with their interaction AB.

Kpr-1SSTTotal

MSE=SSE/kp(r-1)

kp(r-1)SSEError

MSAB/MSEMSAB=SSAB/ (k-1)(p-1)

(k-1)(p-1)SSABInteraction Effect AB

MSB/MSEMSB=SSB/p-1P-1SSBMain Effect B

MSA/MSEMSA=SSA/k-1k-1SSAMain Effect A

F-RatioMean Sum of Squares

dfSum of Squares

Sources of Variation

Kpr-1SSTTotal

MSE=SSE/kp(r-1)

kp(r-1)SSEError

MSAB/MSEMSAB=SSAB/ (k-1)(p-1)

(k-1)(p-1)SSABInteraction Effect AB

MSB/MSEMSB=SSB/p-1P-1SSBMain Effect B

MSA/MSEMSA=SSA/k-1k-1SSAMain Effect A

F-RatioMean Sum of Squares

dfSum of Squares

Sources of Variation

Two-factor with replication - Demo

MS Excel: Put response data for two factors like in a lay out like

in the next page. Select Tools/Data Analysis and select Anova: Two

Factor with replication from the Analysis Tools list. Click OK.

Select Input Range and input the rows per sample: Number of replications (excel needs equal replications for every levels). Replication is 2 for the data in the next page.

Select output range and then click on ok.

Two-factor ANOVA MS-Excel Data Layout:

grp1 grp2 grp3shade1 52.50647 47.83167 43.82211shade1 43.14426 43.69665 40.41359shade1 55.91853 40.73333 57.65748shade1 45.68187 46.56424 41.96207shade1 54.76593 53.03273 49.08051shade1 45.27999 56.41127 55.97797shade1 41.95513 43.69635 47.71419shade1 43.67319 50.92119 41.75912shade1 44.01857 40.36097 46.21859shade1 44.54295 51.95264 53.61966shade2 46.1765 57.1638 49.63484shade2 40.02826 50.98321 57.68229shade2 58.09449 49.23148 49.08471shade2 43.25757 48.01014 40.59866shade2 42.07507 45.98231 38.99055shade2 46.80839 55.32514 54.74286shade2 43.80479 47.21837 53.74624shade2 57.60508 40.30104 44.82507shade2 42.47022 50.56327 45.85581shade2 57.4945 55.55145 41.36863

Two-factor ANOVA MS-Excel output: height on treatment group, shades, and their interaction

Anova: Two-Factor With Replication

SUMMARYgrp1 grp2 grp3 Totalshade1

Count 10 10 10 30Sum 471.4869 475.201 478.2253 1424.913Average 47.14869 47.5201 47.82253 47.49711Variance 26.773 29.80231 38.11298 29.46458

shade2

Count 10 10 10 30Sum 477.8149 500.3302 476.5297 1454.675Average 47.78149 50.03302 47.65297 48.48916Variance 50.87686 26.02977 41.05331 37.84396

Total

Count 20 20 20Sum 949.3017 975.5313 954.7549Average 47.46509 48.77656 47.73775Variance 36.88689 28.10855 37.50739

ANOVASource of Variation SS df MS F P-value F crit

Sample 14.76245 1 14.76245 0.416532 0.521406 4.01954Columns 19.15632 2 9.578159 0.270254 0.764212 3.168246Interaction 18.9572 2 9.478602 0.267445 0.766341 3.168246Within 1913.834 54 35.44137

Total 1966.71 59

Two-factor ANOVA - Demo SPSS:

Select Analyze > General Linear Model > Univariate Make selection of variables e.g. Dependent

varaiable: response (hgt), and Fixed Factor: grp and shades.

Make other selections as follows-click on Post Hoc and select Multiple comparisons (LSD, Tukey, Bonferroni, or Scheffe), click options and select Homogeneity of variance test, click continue and then Ok.

Two-factor ANOVA SPSS output: height on treatment group, shades, and their interaction

Between-Subjects Factors

N 1 20 2 20

grp

3 20 1 30 Shades

2 30

Tests of Between-Subjects Effects Dependent Variable: hgt

Source Type III Sum of Squares df Mean Square F Sig.

Corrected Model 52.876(a) 5 10.575 .298 .912 Intercept 138200.445 1 138200.445 3899.410 .000 grp 19.156 2 9.578 .270 .764 Shades 14.762 1 14.762 .417 .521 grp * Shades 18.957 2 9.479 .267 .766 Error 1913.834 54 35.441 Total 140167.155 60 Corrected Total 1966.710 59

a R Squared = .027 (Adjusted R Squared = -.063)

Two-factor ANOVA R output: height on treatment group, shades, and their interaction> grp<- as.factor(grp)

> shades<-as.factor(shades)

> summary(aov(hgt~grp+shades+grp*shades))

Df Sum Sq Mean Sq F value Pr(>F)

grp 2 19.16 9.58 0.2703 0.7642

shades 1 14.76 14.76 0.4165 0.5214

grp:shades 2 18.96 9.48 0.2674 0.7663

Residuals 54 1913.83 35.44