Last updated on 2024年8月28日

How do you analyze the data from a factorial block design?

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If you want to test the effects of two or more factors on a response variable, you might use a factorial design. But what if your experimental units are not homogeneous or have some inherent variability? In that case, you might need to use a block design to reduce the confounding effects of these nuisance factors. A factorial block design combines the features of both designs, allowing you to partition the sources of variation and estimate the main and interaction effects of your factors. But how do you analyze the data from a factorial block design? Here are some steps to follow.

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Natalie Rodrigue

Skilled Biostatistician & Epidemiologist - Mentor, Experienced, Knowledgeable & Passionate Instructor - DoE - Expert…
Andrew Anderson

IBM Research Intern

1 Choose an appropriate model

The first step is to choose a suitable model for your data. Depending on the number and type of factors and blocks, you might use a fixed, random, or mixed effects model. A fixed effects model assumes that the factor levels and block levels are the only ones of interest and are not randomly selected from a larger population. A random effects model assumes that the factor levels and block levels are randomly selected from a larger population and are not of interest themselves, but rather their variance components. A mixed effects model combines both fixed and random effects in the same model. You should also decide whether you want to include interaction terms or not, depending on your research question and hypothesis.

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Natalie Rodrigue

Skilled Biostatistician & Epidemiologist - Mentor, Experienced, Knowledgeable & Passionate Instructor - DoE - Expert in the Analysis of Genomic Data & Statistical Modelling. + 30 yrs experience.
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The appropriate way to analyse the data depends on the type of blocking structure was used and the size of the experimental units. Whenever there is a single size of experimental units and a blocking structure was used one may use ANOVA (provided that the outcome is continuous and underlying assumptions are used) by incorporating the blocking factors and the other experimental factors and their interactions in the model. For a split-block design, one needs to correctly specify the model for the whole experimental units and the sub experimental units yielding two error terms in the model. One must make sure that the statistical tests rely on the appropriate error terms.

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Andrew Anderson

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ANalysis Of VAriance (ANOVA) is one way to approach an n x m factorial design with continuous dependent variables. One caution is from ANOVA's hypotheses: the null hypothesis states that all means are equal in your n x m experiment. The alternative hypothesis states that AT LEAST ONE of these means is significantly different from the others. A means (hah, stats pun) to find out which one might be: 1) look at a table with your n x m level combinations of means, see which one looks biggest/smallest. 2) visualise your data on a box plot, see which of those means is highest (or lowest). Remove that particular subset of data from analysis, run ANOVA again. if p > .05, that withheld data was responsible. If p < .05, rinse and repeat.

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2 Check the assumptions

The second step is to check the assumptions of your model, such as the normality of the response variable within each combination of factor levels and block levels, the constant variance of the response variable across all combinations, and the independence of observations within and between blocks. To do this, graphical and numerical methods should be used, including histograms, boxplots, QQ-plots, ANOVA tables, residual plots, and tests of normality, homogeneity, and independence.

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Andrew Anderson

IBM Research Intern
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My background is in human subjects work. One critical assumption to check is for independence. If both of your factors of interest are between-subjects, you MUST ensure that a participant features in one (and only one) cell in your table. One assumption that I encourage people to check, which is often overlooked, is for homoscedasticity (the fancy word for equal variance). Visualising your data with the likes of box plots is a good idea, but Levene's test checks this assumption. Its null hypothesis is that all VARIANCES are equal, and its alternative is that AT LEAST ONE is significantly different. Check for p > .05 (one of the few times you'll hope for it). If p < .05, Welch's ANOVA relaxes the assumption of equal variance.

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3 Perform the analysis

The third step is to perform the analysis using your chosen model and software. Depending on the software, you might need to specify the factors, blocks, and interactions as fixed or random effects, and use appropriate commands or functions to fit the model and obtain the output. The output should include estimates of the effects, standard errors, confidence intervals, p-values, and measures of model fit, such as R-squared, adjusted R-squared, or AIC. You should also check for potential outliers, influential points, or multicollinearity issues that might affect the validity of your results.

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4 Interpret the results

The fourth step is to interpret the results of your analysis. You should compare the p-values of the effects with your chosen significance level, usually 0.05, and draw conclusions about the significance of the main and interaction effects of your factors and blocks. You should also report the magnitude and direction of the effects, using estimates, confidence intervals, or effect sizes. You should also comment on the model fit and the assumptions, and discuss any limitations or challenges of your analysis.

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5 Present the results

The fifth step is to present the results of your analysis in a clear and concise way. You should use tables, graphs, or charts to summarize and visualize the effects, the model fit, and the assumptions. You should also use appropriate labels, titles, captions, legends, and scales to make your presentation easy to understand and follow. You should also use plain language and avoid jargon or technical terms when explaining your results to your audience.

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6 Communicate the implications

The final step is to communicate the implications of your results for your research question and hypothesis, and for your field of study or practice. You should relate your results to the existing literature and theory, and highlight how your findings contribute to the knowledge or solve a problem. You should also discuss the limitations and assumptions of your analysis, and suggest directions for future research or action.

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you analyze the data from a factorial block design?

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1 Choose an appropriate model

2 Check the assumptions

3 Perform the analysis

4 Interpret the results

5 Present the results

6 Communicate the implications

7 Here’s what else to consider

Experimental Design

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