9.2 – Chi-square contingency tables

Introduction

This page covers a lot of ground to extend our discussion of implementations of the chi-square test to experimental data and introduces several related statistics, including a correlation coefficient for categorical data and an introduction to agreement and reliability measurement statistics.

We begin with contingency tables.

We just completed a discussion about goodness of fit tests, inferences on categorical traits for which a theoretical distribution or expectation is available. We offered Mendelian ratios as an example in which theory provides clear-cut expectations for the distribution of phenotypes in the F₂ offspring generation. This is an extrinsic model — theory external to the study guides in the calculation of expected values — and it reflects a common analytical task in epidemiology.

But for many other kinds of categorical outcomes, no theory is available. Today, most of us accept the link between tobacco cigarette smoking and lung cancer. The evidence for this link was reported in many studies, some better designed than others.

Consider the famous Doll and Hill (1950) report on smoking and lung cancer. They reported (Table IV Doll and Hill 1950), for men with lung cancer, only two out of 649 were non-smokers. In comparison, 27 case-control patients (ie, patients in the hospital but with other ailments, not lung cancer) were nonsmokers, but 622 were smokers (Table 1).

Table 1. Doll and Hill (1950).

A more recent example from the study of the efficacy of St John’s Wort (Hypericum perforatum) as a treatment for major depression (Shelton et al 2001, Apaydin et al. 2016). Of patients who received St. John’s Wort over 8 weeks, 14 were deemed improved while 98 did not improve. In contrast 5 patients who received the placebo were deemed improved while 102 did not improve (Table 2).

Table 2. St. John’s Wort and depression

These kinds of problems are direct extensions of our risk analysis work in Chapter 7. Now, instead of simply describing the differences between case and control groups by Relative Risk Reduction or odds ratios, we instruct how to do inference on the risk analysis problems.

Thus, faced with absence of coherent theory as guide, the data themselves can be used (intrinsic model), and at least for now, we can employ the test.

In this lesson we will learn how to extend the analyses of categorical data to cases where we do not have a prior expectation — we use the data to generate the tests of hypotheses. This would be an intrinsic model. One of the most common two-variable categorical analysis involves 2X2 contingency tables. We may have one variable that is the treatment variable and the second variable is the outcome variable. Some examples provided in Table 3.

Table 3. Some examples of treatment and outcome variables suitable for contingency table analysis.

Table 4 were examples of published studies returned from a quick PubMed search; there are many examples (meta-analysis opportunities!). However, while they all can be analyzed by contingency tables analysis, they are not exactly the same. In some cases, the treatments are fixed effects, where the researcher selects the levels of the treatments (see Ch12.3 and Ch14.3 for discussion about fixed vs random effects in experimental design).

In other cases, each of these treatment variables need not be actual treatments (in the sense that an experiment was conducted), but it may be easier to think about these as types of experiments. These types of experiments can be distinguished by how the sampling from the reference population were conducted. Before we move on to our main purpose, to discuss how to calculate contingency tables, I wanted to provide some experimental design context by introducing two kinds of sampling (Chapter 5.5).

Our first kind of sampling scheme is called unrestricted sampling. In unrestricted sampling, you collect as many subjects (observations) as possible, then assign subjects to groups. A common approach would be to sample with a grand total in mind; for example, your grant is limited and so you only have enough money to make 1000 copies of your survey and you therefore approach 1000 people. If you categorize the subjects into just two categories (eg, Liberal, Conservative), then you have a binomial sample. If instead you classify the subjects a number of variables, eg, Liberal or Conservative, Education levels, income levels, homeowners or renters, etc., then this approach is called multinomial sampling. The point in either case is that you have just utilized a multinomial sampling approach. The aim is to classify your subjects to their appropriate groups after you have collected the sample.

Logically what must follow unrestricted sampling would be restricted sampling. Sampling would be conducted with one set of “marginal totals” fixed. Margins refers to either the row totals or to the column totals. The sampling scheme is referred to as compound multinomial. The important distinction from the other two types is that the number of individuals in one of the categories is fixed ahead of time. For example, in order to determine if smoking influences respiratory health, you approach as many people as possible to obtain 100 smokers.

The contingency table analysis

We introduced the 2×2 contingency table (Table 4) in Chapter 7.

Table 4. 2X2 contingency table

Contingency tables are all of the form (Table 5)

Table 5.  Basic format of a 2X2 table

Regardless of how the sampling occurred, the analysis of the contingency table is the same; mechanically, we have a chi-square type of problem. In both contingency table and the “goodness of fit” Chi-Square analyses the data types are discrete categories. The difference between GOF and contingency table problems is that we have no theory or external model to tell us about what to expect. Thus, we calculate expected values and the degrees of freedom differently in contingency table problems. To learn about how to perform contingency tables we will work through an example, first the long way, and then using a formula and the a, b, c, and d 2X2 table format. Of course, R can easily do contingency tables calculations for us.

Doctors noticed that a new form of Hepatitis (HG, now referred to as GB virus C), was common in some HIV+ populations (Xiang et al 2001). Accompanying the co-infection, doctors also observed an inverse relationship between HG loads and HIV viral loads: patients with high HG titers often had low HIV levels. Thus, the question was whether co-infection alters the outcome of patients with HIV — do HIV patients co-infected with this HG progress to AIDS and mortality at rates different from non-infected HIV patients? I’ve represented the Xiang et al (2001) data in Table 6.

Table 6. Progression of AIDS for patients co-infected with HG [GB virus C], Xiang et al data (2001)

Our question was: Do HIV patients co-infected with this HG progress to AIDS and mortality at rates different from non-infected HIV patients? A typical approach to analyze such data sets is to view the data as discrete categories and analyze with a contingency table. So we proceed.

Setup the table for analysis

Rules of contingency tables (see Kroonenberg and Verbeek 2018). What follows is a detailed walk through setting up and interpreting a 2X2 table. Note that I maintain our a, b, c, d cell order, with row 1 referencing subjects exposed or part of treatment group and row 2 referencing subjects not exposed (or part of the control group), as introduced in Chapter 7.

The Hepatitis G data from Xiang et al 2001 data, arranged in 2X2 format (Table 7).

Table 7.  Format of 2X2 table Xiang et al (2001) dataset.

The data are placed into the cells labeled a, b, c, and d.

Cell a : The number of HIV+ individuals infected with HG that lived beyond the end of the study.

Cell b : The number of HIV+ individuals infected with HG that died during the study.

Cell c : The number of HIV+ individuals not infected with HG that lived.

Cell d : The number of HIV+ individuals not infected with HG that died.

This is a contingency table because the probability of living or dying for these patients may have been contingent on coinfection with hepatitis G.

State the examined hypotheses

H_O: There is NO association between the probability of living and coinfection with hepatitis G.

For a contingency table this means that there should be the same proportion of individuals with and without hepatitis G that either lived or died (1:1:1:1).

Now, compute the Expected Values in the Four cells (a, b, c, d)

1. Calculate the Expected Proportion of Individuals that lived in Entire Sample:
   Column 1 Total / Total = Expected Proportion of those that lived beyond the study.
2. Calculate Expected Proportion of Individuals that died in Entire Sample:
   Column 2 Total / Total = Expected Proportion that died during the study.
3. Calculate the Expected Proportion of Individuals that lived and were HG+
   Expected Proportion of living (step 1 above) x Row 1 Total = Expected For Cell A.
4. Calculate the Expected Proportion of Individuals that died and were HG+
   Expected Proportion died x Row 1 Total = Expected For Cell B
5. Calculate the Expected Proportion of surviving individuals that were HG-
   Expected Proportion lived x Row 2 Total = Expected For Cell C
6. Calculate the Expected Proportion of individuals that died that were HG-
   Expected Proportion that died x Row 2 Total = Expected For Cell D

Yes, this can get a bit repetitive! But, we are now done — remember, we’re working through this to review how the intrinsic model is applied to obtain expected values.

To summarize what we’ve done so far, we now have

Observations arranged in a 2X2 table

Calculated the expected proportions, ie, the Expected values under the Null Hypothesis.

Now, we proceed to conduct the chi-square test of the null hypothesis — The observed values may differ from these expected values, meaning that the Null Hypothesis is False.

Table 8. Copy Table 6 data set

Get the Expected Values

1. Calculate the Expected Proportion of Individuals that lived in Entire Sample:
   Column 1 Total (198) / Total (362) = 0.5470
2. Calculate Expected Proportion of Individuals that died in Entire Sample:
   Column 2 Total (164) / Total (362) = 0.4530
3. Calculate the Expected Proportion of Individuals that lived and were HG+
   Expected Proportion of living (0.547) x Row 1 Total (218) = Cell A = 119.25
4. Calculate the Expected Proportion of Individuals that died and were HG+
   Expected Proportion died (0.453) x Row 1 Total (218) = Cell B = 98.75.
5. Calculate the Expected Proportion of surviving individuals that were HG-
   Expected Proportion lived (0.547) x Row 2 Total (144) = Cell C = 78.768
6. Calculate the Expected Proportion of individuals that died that were HG-
   Expected Proportion that died 0(.453) X Row 2 Total = Cell D = 65.232

Thus, we have (Table 9)

Table 9. Expected values for Xiang et al (2001) data set (Table 6 and Table 8)

Now we are ready to calculate the Chi-Square Value

Recall the formula for the chi-square test

Table 10. Worked contingency table for Xiang et al (2001) data set (Table 6 and Table 8)

Cell
a	(103 – 87.768)2 = 78.768	7.4547
b	(41 – 65.232)2 = 65.232	9.002
c	(95 – 119.25)2 = 119.25	4.93134
d	(123 – 98.75)2 = 98.75	5.95506
		27.3452

Adding all the parts we have the chi-square test statistic . To proceed with the inference, we test using the chi-square distribution.

Determine the Critical Value of test and evaluate the null hypothesis

Now recall that we can get the critical value of this test in one of two (related!) ways. One, we could run this through our statistical software and get the p-value, the probability of our result and the null hypothesis is true. Second, we look up the critical value from an appropriate statistical table. Here, we present option 2.

We need the Type I error rate α and calculate the degrees of freedom for the problem. By convention, we set α = 0.05. Degrees of freedom for contingency table is calculated as

Degrees of Freedom = (# rows – 1) x (# columns – 1)

and for our example → DF = (2 – 1) x (2 – 1) = 1.

Note. How did we get the 1 degree of freedom? Aren’t there 4 categories and shouldn’t we therefore have k – 1 = 3 df? We do have four categories, yes, but the categories are not independent. We need to take this into account when we evaluate the test and this lack of independence is accounted for by the loss of degrees of freedom.

What exactly is not independent about our study?

1. 1. The total number of observations is set in advance from the data collection.
   2. We also have the column and row totals set in advance. The number of HIV individuals with or without Hepatitis G infection was determined at the beginning of the experiment. The number of individuals that lived or died was also determined before we we conducted the test.
   3. Then the first cell (let us make that cell A) can still vary anywhere from 0 to N _i (sample size of the first drug).
   4. Once the first cell (A) is determined then the next cell in that row (B) will have to add up to N _i.
   5. Also the other cell in the same column as the first cell (C) must add up to the Column 1 Total.
   6. Lastly, the last cell (D) must have the Row 1 Total add up to the correct number.

All this translates into there being only one cell that is FREE TO VARY, hence, only one degree of freedom.

Get the Critical Value from a chi-square distribution table: For our example, look up the critical value for DF = 1, α = 0.05, you should get 3.841. The 3.841 is the value of the chi-square we would expect at 5% and the null hypothesis is in fact true condition in the population.

Once we obtain the critical value we simply use the previous statement regarding the probability of the Null Hypothesis being TRUE. We Reject the Null Hypothesis when the calculated test statistic is greater than the Critical Value. We Accept the Null Hypothesis when the calculated  is less than the Critical Value. In our example we got 27.3452 for our calculated test statistic, thus we reject the null hypothesis and conclude, that, at least for the sample in this study, there was an association between the probability of patients living past the study period and the presence of hepatitis G.

from a, b, c, and d 2X2 table format formula

If a hand calculation is required, a simpler formula to use is

where N is the sum of all observations in the table, r₁ and r₂ are the marginal row totals, and c₁ and c₂ are the marginal column totals from the 2×2 contingency table (eg, Table 4). If you are tempted to try this in a spreadsheet, and assuming your entries for a, b, c, d, and N look like Table 11, then a straightforward interpretation requires referencing five spreadsheet cells no less than 13 times! Not to mention a separate call to the distribution to calculate the p-value (one tailed test).

Table 11. Example spreadsheet with formulas for odds ratio (OR), Pearson’s , and p-value from distribution.

	A	B	C	D	E	F	G
1
2	a	103			OR	=(2*5)/(3*4)
3	b	41
4	c	95
5	d	123			chisq, p	=(((2*5-3*4)^2)*6)/ (SUM(2:3)*SUM(4:5)* SUM(2,4)*SUM(3,5))	=CHIDIST(F5,1)
6	N	632

Now, let’s see how easy this is to do in R.

effect size

Inference on hypotheses about association, there’s statistical significance — evaluate p-value compared to Type I error, eg, 5% — and then there’s biological importance, the strength of the association. The concept of effect size is an attempt to communicate the likely importance of a result. Several statistics are available to communicate effect size: for that’s φ, phi for the φ (phi) coefficient.

where N is the total. Phi for our example is

R code

sqrt(27.3452/362)

and R returns

[1] 0.274844

Effect size statistics typically range from 0 to 1; Cohen (1992) suggested the following interpretation

Table 12. Cohen’s interpretation of d values.

For our example, effect size of the association between co-infection with GB virus C and mortality was weak in the patients with HIV infection. The function phi() in psych package will calculate phi coefficient from a 2X2 table.

Agreement, kappa, and inter-rater reliability

I imagine by now you have made the leap — can we use the contingency table to assess agreement between (inter-) two judges (raters) who evaluate the same object? A naive approach would be to calculate simple agreement, the percent of objects in the list of items evaluated for which both judges agree (concur). For a binomial Yes or No, then agreement (concordance) occurs when both judges report “Yes,” or both report “No,” displayed on the main diagonal of the 2X2 table. The other possible combinations (Yes vs No and No vs Yes) are disagreement (discordance) and would be displayed on the off-diagonal of the 2X2 matrix.

Unfortunately, simple agreement, while intuitive, cannot discriminate between true agreement and agreement by chance. Cohen’s kappa (Cohen 1960, McHugh 2012) is used to assess agreement, aka inter-rater reliability, for binomial and nominal data types for two judges (raters). Cohen’s 1960 paper has been enormously impactful, with more than 53K citations (Google Scholar, Fall 2024), and spawned a number of additional measures of agreement, eg, weighted kappa for more than binomial levels, Fleiss kappa for three or more raters, ICC for ratio scale, discussed in Ch12.3). It’s beyond the scope of this text to review the context of such terms as agreement, and reliability, for just two terms used to evaluate results from two independent observers of the “same” event or object, and leave that to the reader to confirm how to report such endeavors.

Off we go. Cohen’s kappa is defined as

where p_o is the proportion in agreement and p_c is proportion of agreement expected by chance. Essentially, then, interpretable kappa values fall between 0 and 1 (0% to 100%); negative kappa estimates are possible.  Cohen (1960) noted that for kappa to equal 1, the off diagonal elements would all have to be zero.

Cohen provided an estimate of the standard error for kappa as

and 95% confidence intervals are then .

Here, we review calculation of simple agreement and Cohen’s kappa, with just the two levels. Both simple agreement and Cohen’s kappa (κ), are available in R via the icc package. Cohen’s kappa is also available in the psych package, with the cohen.kappa() function providing a confidence interval estimate directly.

Use of both is illustrated here with a simulated coin toss example modified from a nice 2018 summary of Cohen’s kappa at r-bloggers. The R code

library(irr)
# simulate some data
N = 24
flips1 <- rbinom(n = N, size = 1, prob = 0.5)
flips2 <- rbinom(n = N, size = 1, prob = 0.5)
coins<-cbind(flips1, flips2)

Confirm the data set.

head(coins)  
flips1 flips2 
[1,] 0 1 
[2,] 1 0 
[3,] 1 0 
[4,] 1 1 
[5,] 0 0 
[6,] 1 0

and create a 2X2 table.

I stacked the data

head(StackedCoin)
Side Coins    HT
1 0 flips1 tails
2 1 flips1 heads
3 1 flips1 heads
4 1 flips1 heads
5 0 flips1 tails
6 1 flips1 heads

then created a simple 2×2 table (see Note 6 below for a simple explanation for why “.Table” as a named object).

.Table <- xtabs(~Coins+HT, data=StackedCoin)

Table 13. coins as 2X2 table.

We can calculate the phi coefficient for Table 13.

phi(.Table, digits = 3)
[1] -0.167

and would conclude no correlation between the two sets of coin tosses.

Now, agreement statistics. Both agree() and kappa2() expect an unstacked (wide) data format. It’s not necessary, but we’ll call a value of 1 as “heads up” and a value of 0 as “tails up” of the tossed coin.

Next, simple agreement.

agree(coins, tolerance=0)

R returns

Subjects = 24 
Raters = 2 
%-agree = 41.7

Now, a proper agreement statistic.

kappa2(coins) 

and R returns

Subjects = 24 
Kappa = -0.135
z = -0.7 
p-value = 0.484

Results with psych:cohen.kappa, which provides the 95% confidence interval as default output.

Cohen Kappa and Weighted Kappa correlation coefficients and confidence boundaries 
                 lower estimate upper
unweighted kappa -0.51    -0.14  0.24
weighted kappa   -0.51    -0.14  0.24
Number of subjects = 24

By chance, the two independent toss of 24 coins had 10 rows of “agreement” (H:H or T:T), simply due to chance. However, results of kappa find no agreement between the two sets of coin tosses.

In summary, we see that simple agreement and the kappa statistic don’t return similar results. Percent agreement reported 41%, whereas kappa was zero. We know percent agreement can’t reflect anything in the coin toss case because each toss is independent from the next. We’ll have you explore repeated trials of coin tosses to help convince you that simple agreement is not a reliable measure of agreement (see question 6 below).

Interpreting agreement values is not straightforward, and some argue agreement statistics should be avoided in favor of disagreement measures (for example, see discussion in Pontius and Millones 2011). Cohen suggested (Table 14)

Table 14. Landis and Koch’s (1977) interpretation of Cohen’s kappa.

Even a cursory look at Table 14 should lead to questions about scenarios in which a range of agreement “substantial” would provide little comfort about the agreement between the judges or the reliability of the instrument used, for example forensic cases or histology of tissue biopsies.

Contingency table analyses in Rcmdr

Assuming you have already summarized the data, you can enter the data directly in the Rcmdr contingency table form. If your data are not summarized, then you would use Rcmdr's Two-way table… for this. We will proceed under the assumption that you have already obtained the frequencies for each cell in the table.

Rcmdr: Statistics → Contingency tables → Enter and analyze two-way table…

Here you can tell R how many rows and how many columns (Fig 1).

Figure 1. Screenshot R Commander menu for 2X2 data entry with counts.

The default is a 2X2 table. For larger tables, use the sliders to increase the number of rows, columns, or both.

Next, enter the counts. You can edit every cell in this table, including the headers. In the next panel, I will show you the data entry and options. The actual calculation of the test statistic is done when you select the “Chi-square test of independence” check-box (Fig 2).

Figure 2. Display of Xiang et al data entered into R Commander menu.

After entering the data, click on the Statistics tab (Fig 3).

Figure 3. Screenshot Statistics options for contingency table.

If you also select “Components of chi-square statistic” option, then R will show you the contributions of each cell (O-E) towards the chi-square test statistic value. This is helpful to determine if rejection of the null is due to a subset of the categories, and it also forms the basis of the heterogeneity tests, a subject we will pick up in the next section.

Here’s the R output from R Commander. Note the “2, 2, byrow=TRUE” instructions (check out R help to confirm what

.Table <- matrix(c(103,41,95,123), 2, 2, byrow=TRUE)
dimnames(.Table) <- list("rows"=c("HG+", "HG-"), "columns"=c("Lived", "Died"))

.Table # Counts
     columns
rows Lived Died
 HG+   103   41
 HG-   95   123

.Test <- chisq.test(.Table, correct=FALSE)
.Test

Pearson's Chi-squared test

data: .Table
X-squared = 27.339, df = 1, p-value = 0.0000001708

round(.Test$residuals^2, 2) # Chi-square Components
    columns
rows Lived Died
HG+   7.46 9.00
HG-   4.93 5.95

remove(.Test)
remove(.Table)

end R output.

Lots of output, take a deep breath and remember… The minimum output we need to look at is… (Table 15)?

Table 15. Minimally expected outcome for report from a statistical test.

We can see that the p-value 1.7^-7 is much less than Type 1 α = 0.05, thus, by our decision criterion we reject the null hypothesis (provisionally of course, as science goes).

myNumber = 0.0000001708
format(myNumber, scientific = TRUE, digits=2)

[1] "1.7e-07"

Questions

1. Instead of R Commander, try the contingency table problem in R directly.

   myData <- matrix(c(103,41,95,123))
   chisq.test(myData)

   Is this the correct contingency table analysis? Why or why not?

2. For many years National Football League games that ended in ties went to “sudden death,” where the winner was determined by the first score in the extra period of play, regardless of whether or not the other team got an opportunity to possess the ball on offense. Thus, in more than 100 games (140), the team that won the coin toss and therefore got the ball first in overtime, won the game following either a kicked field goal or after a touchdown was scored. In 337 other games under this system, the outcome was not determined by who got the ball first. Many complained that the “sudden death” format was unfair and in 2013 the NFL changed its overtime rules. Beginning 2014 season, both teams got a chance to possess the ball in overtime, unless the team that won the coin toss also went on to score a touchdown at which time that team would be declared the winner. In this new era of over-time rules ten teams that won the coin toss went on to score a touchdown in their first possession and therefore win the game, whereas in 54 other overtime games, the outcome was decided after both teams had a chance on offense (data as of 1 December 2015). These data may be summarized in the table
   

   	First possession win?
   	Yes	No
   Coin flip years	140	337
   New era	10	54

   A. What is the null hypothesis?
   B. Which is more appropriate: to calculate an odds ratio or to calculate an RRR?
   C. This is a contingency table problem. Explain why
   D. Conduct the test of the null hypothesis.
   E. What is the value of the test statistic? Degrees of freedom? P-value?
   F. Evaluate the results of your analysis — do you accept or reject the null hypothesis?

3. Return to the Doll and Hill (1950) data: 2 men with lung cancer were nonsmokers, 647 men with lung cancer were cigarette smokers. In comparison, 27 case-control patients (ie, patients in the hospital but with other ailments, not lung cancer) were nonsmokers, but 622 were cigarette smokers.
   A. What is the null hypothesis?
   B. Which is more appropriate: to calculate an odds ratio or to calculate an RRR?
   C. This is a contingency table problem. Explain why
   D. Conduct the test of the null hypothesis.
   E. What is the value of the test statistic? Degrees of freedom? P-value?
   F. Evaluate the results of your analysis — do you accept or reject the null hypothesis?
4. A more recent example from the study of the efficacy of St John’s Wort as a treatment for major depression (Shelton et al 2001). Of patients who received St. John’s Wort over 8 weeks, 14 were deemed improved while 98 did not improve. In contrast 5 patients who received the placebo were deemed improved while 102 did not improve.
   A. What is the null hypothesis?
   B. Which is more appropriate: to calculate an odds ratio or to calculate an RRR?
   C. This is a contingency table problem. Explain why
   D. Conduct the test of the null hypothesis.
   E. What is the value of the test statistic? Degrees of freedom? P-value?
   F. Evaluate the results of your analysis — do you accept or reject the null hypothesis?
5. Return to the coin toss simulation and repeat comparisons between simple agreement and Cohen’s kappa for different values of N. Do you see any trend(s) for the two measures as sample size increases or decreases?
6. Repeat the coin toss simulation for N = 24 up to five times, each time calculate simple agreement and Cohen’s kappa. Provide a table of your five results plus the example, and generalize simple agreement vs Cohen’s kappa as a reliable measure of agreement. Hint: you may want to use Descriptive statistics to help make your case.

Quiz Chapter 9.2

Chi-square contingency tables

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Chapter 9 contents

• Introduction
• Chi-square test: Goodness of fit
• Chi-square contingency tables
• Yates continuity correction
• Heterogeneity chi-square tests
• Fisher exact test
• McNemar’s test
• References and suggested readings

5 – Experimental design

Beyond numbers.

During this course, you will learn about statistics, yes, but my hope and goal for your experience in this class is much more than that: statistical reasoning. To have statistical reasoning skills, you need to be comfortable with the context of how data are acquired, ie, data acquisition. It’s not just about knowing how an instrument performs, eg, under or over the limits of quantification characteristics of the instrument lead to censored missing values, missing not at random (MNAR). In a broad stoke view, data are obtained from two kinds of studies: observational studies and experimental (manipulative) studies. what a correctly designed experiment can tell you about the world, and how a poorly designed experiment works against you. At the end of the semester, you should be familiar with the issues of Randomization, Control, Independence, and Replication.

The basics explained.

Experimental design is a discipline within statistics concerned with the analysis and design of experiments. Design is intended to help research create experiments such that cause and effect can be established from tests of the hypothesis. We introduced elements of experimental design in Chapter 2.4. I recommend a quick re-read: We talk about the different types of bias that can affect research. We also point out parts of the problem you’ll need to think about when planning a project, so you can avoid or reduce bias from the start.

Here, we expand our discussion of experimental design. We begin with review of definitions and an outline of an important concept in statistics. First, we need to be clear about why we measure or conduct experiments. We do so to collect data (datum is the singular) about a characteristic or trait from a population. Data are observations: they include observations and measurements from instruments. At their best, data are the “facts” of science.

Measurement, Observation, Variables, Values.

Measurement is how we as scientists acquire our data (Chapter 3.4). The process of measuring involves the assignment of numbers, codes, or labels to observations according to rules established prior to any data collection (Stevens 1946, Houle et al 2011). Observations refer to the units of measurement, whereas variables are the characteristics or traits that are measured. A value refers to the particular number, score, or label assigned to a particular sample for a variable. Variables are generic, the thing being measured, values are specific to the subject or sample being measured for the variable. Each discipline in biology has its own set of variables and samples may or may not have different values for each variable measured. Variables are summarized as a statistic (eg, the sample mean), which is a number taken to estimate a parameter, which pertains to the population. Variables and parameters in statistics were discussed in Chapter 3.4. Because numbers or scores or labels can be assigned according to different rules, this means that variables may be measured on different kinds of scales or data types. The different kinds of data types were presented in Chapter 3.1.

Missing values.

A missing value refers to lack of a value for an observation or variable (Chapter 3.5).  Censored values, a type of missing value, include observations for which limited or partial information is available, eg, range, while a missing value lacks any information about the true value.

Missing values can affect analysis and many R algorithms are sensitive or may fail to run in the presence of missing values. Missing data may be of three kinds, and one of them, missing not at random (MNAR), can substantially influence the analysis. MNAR implies some observations are missing because of a systematic bias. Instrument limits of quantification are an example of systematic bias — for example, spectrophotometric absorbance readings of zero for a colorimetric assay (eg, Bradford protein assay) may not represent complete absence of the target, but rather, the lower detection limits of the instrument or the assay method (0.1 mg protein/mL in this case).

The other kinds of missing values are missing completely at random (MCAR) and missing at random (MAR). MCAR implies there is no association between any element of the experiment and the absence of a value. Analysis on MCAR data sets may support unbiased conclusions. MAR includes the random errors that occur during data acquisition: data may be lost because of operator error or power outage (shudder). Unlike MNAR, analysis of MAR, like MCAR data sets may still result in unbiased conclusions.

One consequence of missing values — a decision how best to handle the lost information. The options are either pairwise deletion or listwise deletion. Listwise means the entire row, all variables, are lost for any record with one or more missing values. You recorded root length, stem length, numbers of secondary roots, time to germinate, but the SPAD meter failed and there is no chlorophyl reading for seedling #212? Sorry, the entire row is deleted. Optionally, the missing data is deleted pairwise — only chlorophyl reading for #212 is lost, all other variables remain.

Users of R will quickly learn some algorithms immediately fail to run if not instructed how to process missing values. In some cases, missing values can be replaced by imputed values.

Cause & Effect.

Observations or measurements gathered under controlled conditions (experiments) are essential if we are to answer questions about populations, to separate cause and effect, where one or more events is directly the result of another factor, as opposed to anecdote, a story about an event which, by itself, cannot be used to cause from association (eg, spurious correlations, see Ch 16.2). Recall that anecdotal evidence typically comes from personal experience, where observations may be obtained by non-systematic methods. Well designed experiments, in the classical sense, permit discrimination among competing hypotheses in large part because observations are collected according to strict rules. Well developed hypotheses tested by well-designed experiments permit ruling out alternative explanations (sensu Pratt [1964] strong inference). Observational studies, or epidemiology studies if we are talking about investigations of risk assessment, also may contribute to discussions of cause and effect (see history of smoking by Doll 1998).

Medicine is replete with stories about how a patient showed a particular set of symptoms, and how a physician applied a set of diagnostic protocols. An outcome was achieved, and the physician reports the outcome and circumstances related to the patient to her colleagues. This is an example of a case study, and the focus of investigation is the individual, the patient. The doctor’s report will sound like, “I tried the standard treatment given the diagnosis, and the patient’s symptoms diminished, but later returned. I tried a higher dose, but the patient’s symptoms persisted unabated.” No inferences are made to a wider set of individuals and the report is anecdotal. If observations are made on several patients, this may be a case series.

In ecology, a field biologist may notice a six-legged adult frog (Scott 1999, Alvarez et al 2021) — since frogs typically have four legs, the six-legged frog attracts the biologist’s eye and he jots down the circumstances in which the frog were found: relative humidity, air temperature, ground temperature, where the frogs was found (on lower leaf of a philodendron plant). A water source near where the frog was found is tested for pH with a meter the biologist carries, and a water sample is taken for later testing of herbicides. The frog is collected so that it can be checked for skin parasites. Upon further inspection, the frog did indeed have parasites known to cause deformities in other frog species. However, note that this example too, is a case study. Although the biologist makes additional observations, any conclusions about why the frog has six legs is anecdotal.

From these case studies, no conclusions can be drawn. We cannot say why the patient failed to respond to treatment, nor can we say why the frog has six legs. Why? Because these are singular events, and a variety of explanations can be given as to their causes — importantly, no controls are available, so there’s no way to distinguish among possibilities.

From such anecdotes, however, experiments can be designed. The physician may decide to recruit additional patients with the diagnosed illness and apply the standard treatment to see if her anecdote is a single, unique event, or more indicative of a problem with the treatment. The biologist may collect other frogs from the area near where he found the six-legged frog and check to see if they, too, have the parasites. If additional patients fail to respond to the treatment, then the singular even is more likely to be a phenomenon. If the normal frogs also have similar levels of the parasite then it is unlikely that these parasites caused the malformed frog. With this simple step (recruiting similar patients, finding additional frogs), we can begin to make inferences about cause and effect and in some cases, to generalize our findings.

This is the objective of most statistical procedures, the concept of sampling from a reference population and making distinctions between groups within the sample. The difference between observational and experimental studies then is how the subjects are selected with respect to the groups. In an experiment, the researcher controls and decides which subject receives the treatment, therefore, allocation to groups is manipulated by the researcher. In contrast, subjects included in an observational study have already been “assigned” to a group, but not by us. Assignment to groups, smokers or non smokers, Type II diabetes or no diabetes (etc.), is done by nature.

Now, I do not wish to imply that research that cannot be generalized back to a reference population are worthless. Far from it. In fact, there is a strong argument for specificity. For example, much basic biological research depends on work or model organisms, which in turn may be further partitioned into specific genetic lines (cf. discussion in Rothman et al 2013). And my goodness, what we have learned about the devastation to oceanic islands like Guam when the brown tree snake was introduced (Fritts and Rodda 1998).  Strictly speaking, what has happened to Guam is a case history. But no one would argue that what has happened to Guam cannot happen to Hawaii and other oceanic islands (eg, United States Federal law 384-108 “Brown Tree Snake Control and Eradication Act of 2004”). In other words, even from case histories, generalizations can sometimes be made.

There can also be real reasons to ignore the issue of generality. One benefit of specificity is experimental control. Transgenic lines may differ by single gene knockout or by gene duplication, and clearly the aim of such studies is to evaluate the function (hence purpose) of the gene product (or its absence) on some phenotype. In this sense it may not seem important that an transgenic mouse is not representative of a wild outbred mouse population. However, this argument is fundamentally one of expedience — such studies do result in specific results, results that cannot be generalized beyond the strains involved. It ignores the issue of genetic background — all of the genes that affect a trait in addition to the candidate gene under study (Sigmund 2000; Lariviere et al 2001). Transgenic mice of different inbred strains or their hybrids may have very different alleles at other genes that may influence a phenotype, hence, the results of the gene knockdown or other engineering result in different outcomes. Results of genetic manipulations on inbred strains, no matter how sophisticated, the conclusions are strain- or hybrid cross-specific. Thus, although technically and financially difficult, conclusions are better, more generalizable, when conducted with many different inbred lines and verified in outbred mouse populations precisely because genetic background often influences function of single genes (Sigmund 2000; Lariviere et al 2001).

“Causal criteria:” Logic of causation in medicine.

This section is in progress. Just a list of key points and references

Throughout this text, emphasis on the power of experimentation is emphasized. Well designed experiments …

Henle-Koch’s postulates (1877, 1882), working on tuberculosis reported a set of causal criteria to establish link between a microorganism and a disease, are the following:

• The microorganism must be found in abundance in all organisms suffering from the disease, but should not be found in healthy organisms.
• The microorganism must be isolated from a diseased organism and grown in pure culture.
• The cultured microorganism should cause disease when introduced into a healthy organism.
• The microorganism must be reisolated from the inoculated, diseased experimental host and identified as being identical to the original specific causative agent.

Robert Koch wrote these more than 100 years ago, so, clearly, understanding of infectious disease has improved. Evan’s postulates, quoted from A Dictionary of Epidemiology, 5th edition (pp. 86-87).

1. Prevalence of the disease should be significantly higher in those exposed to the hypothesized cause than in controls not so exposed.
2. Exposure to the hypothesized cause should be more frequent among those with the disease than in controls without the disease—when all other risk factors are held constant.
3. Incidence of the disease should be significantly higher in those exposed to the hypothesized cause than in those not so exposed, as shown by prospective studies.
4. The disease should follow exposure to the hypothesized causative agent with a normal or log-normal distribution of incubation periods.
5. A spectrum of host responses should follow exposure to the hypothesized agent along a logical biological gradient from mild to severe.
6. A measurable host response following exposure to the hypothesized cause should have a high probability of appearing in those lacking this before exposure (eg, antibody, cancer cells) or should increase in magnitude if present before exposure. This response pattern should occur infrequently in persons not so exposed.
7. Experimental reproduction of the disease should occur more frequently in animals or humans appropriately exposed to the hypothesized cause than in those not so exposed; this exposure may be deliberate in volunteers, experimentally induced in the laboratory, or may represent a regulation of natural exposure.
8. Elimination or modification of the hypothesized cause should decrease the incidence of the disease (eg, attenuation of a virus, removal of tar from cigarettes).
9. Prevention or modification of the host’s response on exposure to the hypothesized cause should decrease or eliminate the disease (eg, immunization, drugs to lower cholesterol, specific lymphocyte transfer factor in cancer).
10. All of the relationships and findings should make biological and epidemiological sense.

Fredericks, D. N., & Relman, D. A. (1996). Sequence-based identification of microbial pathogens: a reconsideration of Koch’s postulates. Clinical microbiology reviews, 9(1), 18-33.

Correlation (association) does not imply causation, a well-worn truism in any application of critical thinking skills. Conversely, causation does imply association (correlation); If a change in one variable directly changes another variable, there will be a statistical association between the two variables. Thus, a correlation between two variables is necessary, but not sufficient evidence for cause and effect between two variables.

However, sometimes association does point to a cause. A familiar example is association between tobacco cigarette smoking causes lung cancer.  Surgeon General Luthar Terry’s 1964 report (link to document in National Library of Medicine), presented a strong case linking smoking to elevated risk of lung cancer and coronary artery disease.

Bradford Hill’s guidelines to evaluate causal effects based on epidemiology (Hill 1965, see also Sussar 1999, Fedak et al 2015). Set of necessary and sufficient conditions, inductive reasoning.

1. Strength of association
2. Consistency of observed association
3. Specificity of association
4. Temporal relationship of the association
5. Biological gradient, eg, a dose-response curve
6. Biological plausibility
7. Coherence, the cause and effect inference should not conflict with what is known about the etiology of a disease.

Follows and extends David Hume’s (1739) causation criteria: association (Hill #1), cause precedes effect (Hill #4), direction of connection.

Validity.

An obvious objective of research is to reach valid conclusions about fundamental questions. A helpful distinction between the specific and the generalizable experiment is to recognize there are two forms of validity in research: internal validity and external validity (Elwood 2013). Internal validity is the quality of a designed study that determines whether cause and effect can be determined. Random assignment of subjects to treatment groups enhances the internal validity of the study. External validity relates to how general the assessment of cause and effect can be to other populations. Thus, random sampling from a reference population has to do with whether or not the study has external validity.

Additional definitions.

We proceed now with definitions. We use the term population in a special and restrictive way in statistics. Our definition includes the one you are already familiar with, but it also means more than that.

Populations are the entire group of individuals that you want to investigate. In statistics, the entire groups is actually the entire class with the observation — so if we are referring to the average body weight of house mice, we’re actually referring to the body weight as the population — it’s a subtle distinction, not essential for our introduction to biostatistics. When we conduct experiments and apply statistical tests on collected data, we generally intend to make inferences (draw conclusions) from our results back to the population.

Population has a strict application in statistics, but the definition also includes our general understanding of the word population. For example, Examples of a population in the general sense one may refer to include: .… the entire human population existing today .… the entire collection of U.S. citizens. .… all the individuals in an entire species. .… all individuals in a population of a species (eg, house mice in a dairy barn in Hawai’i) .… all of us in this classroom (if we are ONLY interested in US) If you could measure the entire population then there would be no need to do (or learn) statistics! Populations usually are in the thousands, millions, or billions of individuals. Here, population is used in the everyday sense that we think of — a collection of individuals that share a characteristic.

A more formal definition of “population” in statistics reads as follows: A statistical population is the complete set of possible measurements on a trait or characteristic corresponding to the entire collection of sampling units for which conclusions are intended.

To conclude, in this class, when we talk about population, we will generally be using it in the everyday sense of the word. However, keep in mind that the definition is more restrictive than that and the key is to identify what sampling units are measured.

Conclusions.

This is only the beginning, the basics of experimental design. Beginning with RA Fisher‘s 1935 book, The Design of Experiments (nine editions, some freely accessible via internet search), which is still very much worth reading, entire books are written on the subject, as you can well imagine.

We will also return to the subject of Experimental Design throughout the book. We will return to random sampling in Chapter 5.5. Next we discuss distinctions between experiments and observational studies with respect to sampling of populations.

A bit of a disclaimer here before proceeding; while I cite several papers for examples in experimental design in Chapter 5, reader’s should not read into this that I am either criticizing or endorsing the published experiments. Experimental design will always have elements of compromise — the trick of course is knowing which choices influence validity (Thompson and Panacek 2006).

Questions.

1. Define in your own words the following terms

• reference population
• subjects
• specific versus general conclusions
• random sampling
• convenience sampling
• haphazard sampling
• research validity

2. Revisit our cell experiment, “What is the sampling unit in the following cell experiment?” How would you change this experiment so that there will be biological and not just technical replication?
3. Describe the type of sampling for each research scenario described?

• All African snails on a staircase at Chaminade University are collected on a Thursday evening.
• All African snails on a staircase at Chaminade University are collected every Thursday evening for six months.
• All African snails on all staircases at Chaminade University are collected.
• African snails are studied in the lab, then returned to the areas from which they were collected. Days later, the researcher collects snails from the same area.
• African snails are studied in the lab, then returned to the areas from which they were collected. Days later, the researcher collects snails from a different area.
• The Chaminade University campus is divided into grids. Grids that include stairwells are marked. Before collecting snails, the researcher randomly selects from the list of grids and searches for snails only in those grids selected from the list.
• 
  Figure 1. Giant African Snail (Lissachatina fulica, formerly Achatina fulica). Image by M. Dohm.

4. This next scenario may be evaluated by you for potential sources of bias. Review the list of bias listed above. A researcher wants to do a population count of feral cats on campus. Feral cats are active at night, so he decides to set up a feeding station near a light post. The researcher sits all night in a parked car yards away and watches the feeding station for visits by cats. The researcher repeats these observations over the course of a week, moving the feeding station to different campus locations each night, and reports the total number of cats seen during the week as an estimate of the population size. Be able to discuss this study in terms of potential and actual bias.

Quiz

Including the quiz on this page, a total of 65 questions among the several subchapters, a mix of true or false and multiple choice question format.

Quiz Chapter 5

Experimental design

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Chapter 5 contents

• The basics explained
• Experiments
• Experimental and Sampling units
• Replication, Bias, and Nuisance Variables
• Clinical trials
• Importance of randomization in experimental design
• Sampling from Populations
• References and suggested readings