Biostatistics-Lecture 9Experimental designs

Ruibin XiPeking University

School of Mathematical Sciences

Two-way ANOVA

• A study investigated the effects of 4 treatments (A, B, C, D) on 3 toxic agents (I, II, III).

• 48 rats were randomly assigned to 12 factor level combinations.

Two-way ANOVA

• Y: the response variable• Factor A with levels i=1 to a• Factor B with levels j = 1 to b• A particular combination of levels is called a

treatment or a cell. There are treatments• is the kth observation for treatment (i,j), k

= 1 to n kjiY ,,

ab

Two-way ANOVA

Two-way ANOVA

Two-way ANOVA

Two-way ANOVA

• The factor effect model

Two-way ANOVA

• The factor effect model: constraints

Interaction plots

Two-way ANOVA

• Estimates

• Sum of squares (SS)

Two-way ANOVA

Two-way ANOVA

• Test for Factor A

• Test For Factor B

Two-way ANOVA

• Test for Interaction Effect

Two-way ANOVA

• One observation per cell (n=1)– Cannot estimate the interaction, have to assume

no interaction

Randomized Complete Block Design

• Useful when the experiments are non-homogenous– Rats are bred from different labs– Patients belong to different age groups

• Randomized Block design can used to reduce the variance

Randomized Complete Block Design

Randomized Complete Block Design

• A “block” consists of a complete replication of the set of treatments

• Block and treatments usually are assumed not having interactions

• Advantages:– Effective grouping can give substantially more precise results– Can accommodate any number of treatments and replications– Statistical analysis is relatively simple– If an entire block needs to be dropped, the analysis is not

complicated thereby

Randomized Complete Block Design

• Disadvantages– The degree of freedom for experiment error are not

as large as with a completely randomized design– More assumptions (no interaction between block and

treatment, constant variance from block to block)– Blocking is an observational factor and not an

experimental factor, cause-and-effect inferences cannot be made for the blocking variable and the response

Randomized Complete Block Design

• The model (similar to additive two-way ANOVA)

• block effect treatment effect

Random effect designs

• Fixed effect models– Levels of each factor are fixed– Interested in differences in response among those

specific levels• Random effect model– Random effect factor: factor levels are meant to be

representative of a general population of possible levels

• If there are both fixed and random effects, call it mixed effect model

Random effect designs

• One way random model

Random effect designs

Random Effect designs

Random Effect designs

• Hypothesis

• Testing statistic

Random effect designs

• Two random factors

Random effect designs

• There are five parameters in this model

• For balanced design

Random effect designs

• Hypothesis testing: main effects

Random effect designs

• Hypothesis testing: interaction effects

To be continued

• Mixed effect models• Unbalanced two-way ANOVA