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. An Experimental Unit is the entity on which measurement or an observation is made. For example, subjects are experimental units in most studies. - PowerPoint PPT Presentation
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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.
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
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ij
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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:
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
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ithα
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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 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.