10 – Inferences, Quantitative: Two Sample tests

Testing differences in averages: From one group to two

A one sample parametric test compares the mean against a population value. The population value may come literally from census information, or, more likely, it comes from some applicable theory. The one sample t-test was presented along with how to calculate the confidence interval in the previous chapter.

In this chapter, we also extend to consider two samples, tests about hypotheses for two groups. The two groups may consist of observations on different subjects, and thus the two groups are independent of each other — an independent sample t-test may be used to test null hypothesis. This is called a crossed design or between-subjects design –where each group has its own unique set of observations, and no relationship exists between the observations in the two groups. We discuss crossed experimental designs in Chapter 14.1.

A common experimental design is to measure individuals two or more times, eg, observations like body mass index, BMI, recorded on individuals at the start of an exercise program, and again on the same individuals some time after a treatment — a repeated measures design. In this case, the measures are paired and are, thus, not independent, and a paired-sample t-test would be advised. If the repeated measures occur at different times, then the design is also longitudinal design or within-subjects design. Paired data are examples of nested design — the data points are clustered within individual units. We discuss nested experimental designs in Chapter 14.5.

Two sample parametric tests are used to answer questions about the mean where the data are collected from two random samples of independent observations, each from an underlying normal distribution. The samples may be independent or paired, in which different hypotheses are tested.

Suggested homework

Homework 7: t-tests and ANOVA

Quizzes in this chapter

A total of 27 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 10 contents

• Introduction
• Compare two independent sample means
• Digging deeper into t-test Plus the Welch test
• Paired t-test
• References and suggested reading

5.1 – Experiments

How to put an experiment together

With the background behind us, we outline a designed experiment. Readers may wish to review concepts presented previously, in

• Chapter 2.4: Experimental design and rise of statistics in medical research.
• Chapter 3.4 – Estimating parameters
• Chapter 3.5 – Statistics of error

Experiments have the following elements

1. Define a reference population.
   1. All patients with similar angina symptoms — shortness of breath, chest pain, dizziness — but who have not suffered a heart attack, or myocardial infarction (Wu et al 2017);
   2. All adult frogs in a pond near industrial waste runoff (eg, Sowers et al 2009).
2. Define sample unit from the reference population (eg, as many patients as can be screened by the physician are recruited; all frogs visible to the biologist are captured).
3. Design sampling scheme from population (eg, random sampling vs. convenience/haphazard sampling).
4. Agree on the primary outcome or endpoint to be measured and whether additional secondary outcomes are to be observed and collected.
   1. For the patients with similar symptoms, we follow
   2. For the frogs, we count adults with developmental malformations.
5. Separate the sample into groups so that comparisons can be made
   1. We may be tempted to provide the same treatment and follow the group for improvement — that is the clinically valid approach; to be an experiment, we would use random assignment of individuals to one of two groups — half will receive a placebo, half will receive a beta blocker;,
   2. The frog example doesn’t follow this scheme — the event has already occured, it’s an observational study.
6. Devise a way to exclude or distinguish from the possible explanations, or alternative hypotheses.
   1. eg, for the physician, she will record how many of her patients fail to respond to treatment;
   2. for the biologist, noting presence or absence of parasites between the normal and abnormal frog defines the groups).

A basic experimental design looks like this.

1. One treatment, with two levels (eg, a control group and an experimental group)
2. A collection of individuals recruited from a defined population, esp. by random sampling.
3. Random assignment of multiple individuals to one of the two treatment levels.
4. The treatments are applied to each individual in the study.
5. A measure of response (the primary outcome) and additional features (secondary outcomes) are recorded for each individual.

Of course, there are other considerations, and by now, students reading this text should be able to identify what is missing from the list. (See quiz questions.)

Contrast with design of observational studies

Importantly, for observational studies, no matter how sophisticated the equipment used to measure the outcome variable(s), key steps outlined above are missing. Researchers conducting observational studies do not control allocation of subjects to treatment groups (steps 4 and 3 in the two above lists, respectively). Instead, they may use case-control approaches, where individuals with (case) and without (control) the condition (eg, lung cancer) are compared with respect to some candidate risk factor (eg, smoking). Both case and control groups likely have members who smoke, but if there is an association between smoking and lung cancer, then more smokers will be in the case group compared to the control group.

A cohort study, also a form of observational study, is similar to case-control except that the outcome status is not known. A cohort study includes, in our example, smokers and non smokers who share other characteristics: age, medical history, etc. The other kind of design you will see is the cross-sectional study. Cross-sectional studies are descriptive studies, and, therefore also observational. Primary outcomes along with additional characteristics and outcomes are measured for a representative subsample, or perhaps even an entire population. Cross-sectional studies are used to absolute and relative risk rates. In ecology and evolutionary biology cross sectional studies are common, eg, comparisons of species for metabolic rate (Darveau et al 2005) or life history traits (Jennings et al 1998), and specialized statistical approaches that incorporate phylogenetic information about the species are now the hallmark of these kinds of studies.

A word on outcomes of an experiment. Experiments should be designed to address an important question. The outcomes the researcher measures should be directly related to the important question. Thus, in the design of clinical trials, researchers distinguish between primary and secondary outcomes or endpoints. In educational research the primary question is whether or not students exposed to different teaching styles (eg, lecture-style or active-learning approaches) score higher on an knowledge-content exams. The primary outcome would be the scores on the exams; many possible secondary outcomes might be collected including student’s attitudes towards the subject or perceptions on how much they have learned.

About Primary and Secondary Outcomes.

Consider a evolutionary physiology project on species differences and effects of body size on burst running speed of terrestrial lizards (eg, Bauwens et al 1995; Dohm et al 1998; Labonte et al 2024). Our independent variables, the things we believe influence running performance — and therefore we wish to control for — species, body size, core body temperature, body condition, race quality (see cited papers for these and other examples).

What do we measure? What are the likely primary vs. secondary outcomes for a this kind of project?

Primary outcomes are the direct, definitive measures of maximal locomotor performance. Examples may include

• Maximum Burst Speed: The absolute highest velocity achieved by the lizard over a short distance or time interval (eg, across a 0.5-meter interval).
• Maximum Acceleration: The highest rate of speed change during the start of the burst or during a sudden sprint increase.
• Stride Length: The physical distance covered during a single full stride cycle at maximum speed.
• Stride Frequency: The number of strides taken per second at maximum speed.

And what other measures may be of interest — secondary outcomes — those that may help explain why or how the primary performance metrics change across the independent variables. To list a few,

• Stance and Swing Duration: The individual time durations spent with feet on the ground (stance) vs. in the air (swing).
• Gait Transition Thresholds: The speed at which a lizard shifts from a walk/trot to a high-speed gallop or bipedal running (if applicable to the species).
• Fatigue Rate: The decline in burst speed across successive trials or over a sustained treadmill run.
• Trial-to-Trial Repeatability: The statistical consistency (intraclass correlation coefficient) of a lizard’s performance across multiple bursts.

In a prospective cohort study, we may track participants over time to observe how baseline factors predict the development or worsening of type 2 diabetes (eg, Gao et al 2026). Progression over a two year followup would be measured by incident clinical events or the longitudinal rate of change in biomarkers. Possible primary outcomes are

• Glycated Hemoglobin (HbA1c): A biomarker reflecting average blood glucose levels over the past 2 to 3 months (glycemic control).
• Fasting Plasma Glucose (FPG): Glucose concentration in the blood after an overnight fast (acute baseline glycemic regulation).
• T2D Complication Stage: A clinical score or categorization indicating the presence and severity of microvascular or macrovascular damage.

Secondary outcomes may include

• Fasting Lipid Profile
• C-Reactive Protein
• Body Mass Index
• Estimated Glomerular Filtration Rate

If one or more primary outcomes are significant, this reveals whether an intervention or exposure (like a vegan diet or income level) changes the clinical course of Type 2 Diabetes. Secondary outcomes are critical for explaining how and why it happens. Secondary outcomes provide mechanistic context.

Some caution is warranted — imagine if, after conducting the study, we find diet, age, or income did not statistically affect your primary outcome, but did significantly alter a secondary outcome (eg, improved lipid profiles). What to report? We state in your Abstract and Discussion that the study failed to meet its primary objective, and that any significant secondary findings are exploratory. We cannot claim that a diet “successfully manages diabetes progression” based solely on a secondary marker (cf discussion and suggsted guidelines in Butcher et al 2022).

Example of an experiment.

We’ll work through a familiar example. The researcher is given the task to design a study to test the efficacy in reducing tension headache symptoms by a new pain reliever. There are many possible outcomes: blurry vision, duration, frequency, nausea, need for and response to pain medication, and level of severity (Mayo Clinic). The drug is packaged in a capsule and a placebo is designed that contains all ingredients except the new drug. Forty subjects with headaches are randomly selected from a population of headache sufferers. All forty subjects sign the consent form and agree to be part of the study. The subjects also are informed that while they are participating in the research study on a new pain reliever, each subject has a 50-50 chance of receiving a placebo and not the new drug. The researcher then randomly assigns twenty of the subjects to receive the drug treatment and twenty to receive the placebo, places either a placebo pill of the treatment pill into a numbered envelope and gives the envelopes to a research partner. The partner then gives the envelopes to the patients. Both patients and the research partner are kept ignorant of the assignment to treatment.

We can summarize this most basic experimental design in a table, Table 1.

Table 1. Simple formulation of a 2X2 experiment (aka 2×2 contingency table).

where N is the total number of subjects, a is number of subjects who DID receive the treatment AND got better, c is number of subjects who DID NOT receive the treatment, but DID get better, b is number of subjects who DID receive the treatment and DID NOT get better, and d is number of subjects who DID NOT receive the treatment and DID NOT get better.

And our basic expectation is that we are testing whether or not treatment levels were associated with subjects “getting better” as measured on some scale.

One possible result of the experiment, although unlikely, all of the negative outcomes are found in the group that did not receive the treatment (Table 2).

Table 2. One possible outcome of our 2X2 experiment.

Results of an experiment probably won’t be as clear as in Table 2. Treatments may be effective, but not everyone benefits. Thus, results like Table 3 may be more typical.

Table 3. A more likely outcome of our 2X2 experiment.

Must a control group always be included in an experiment?

To be an experiment, yes. However, it is not true that the control treatment must only be applied to separate, individual subjects. For example, a common experimental design would be repeated measures of an outcome, like behavior (eg, feeding response) before and after an intervention is applied. Thus, the individual is its own control or reference point. For example, Dohm et al (2008) measured feeding and locomotor behavior of adult toads (Rhinella marina, formerly Bufo marinus) prior to and after ozone exposure. We concluded that a single 4-h exposure to O₃ depressed toad feeding behavior, but had little effect on voluntary locomotor behavior. Typically, the design randomly assigns individuals to the  treatments — some individuals get the exposure first, then the control exposure second; others, again selected at random, receive the control exposure first, then the active exposure.

The within subjects design is an important method in biological research; these designs offer control for individual differences as confounding variables and require fewer participants compared to separate control group designs. This design is effective provided there is little risk of carryover effects; if the effects of a previous condition linger, then it may influence a participant’s response in subsequent conditions, confounding the results. For our study (Dohm et al 2008) we used a modified within-subjects design by employing a split plot design. We return to within-subjects designs in Chapter 14.4 – Randomized block design.

Questions

1. When I updated this page in 2020, we were in our fifth month of the coronavirus pandemic of 2019-2020. Daily it seemed, we heard news about “promising coronavirus treatments,” but no study had at the time been published that would meets our considerations for a proper experiment. However, on May 1, 2020, the FDA issued an emergency use authorization for use of the antiviral drug remdesivir (Gilead), based on early results of clinical trials reported 29 April in The Lancet (Wang et al 2020). Briefly, their study included 158 treated with does of the antiviral drug and 79 provided with a placebo control. Both groups were treated otherwise the same. Improvement over 28 days was recorded: 62 improved with Placebo and 133 improved receiving dose of remdesivir.
   1. Using the background described on this page, list the information needed to design a proper experiment. Using your list, review the work described in The Lancet article and check for evidence that the trial meets these requirements.
   2. Create a 2×2 table with the described results from The Lancet article.
2. Consider our hazel tea and copper solution experiment described in Chapter 3. The outcome variables (Tail length, tail percentage, Olive moment) are quantitative, not categorical. Create a table to display the experimental design.
3. For the migraine example, identify elements of the design that conform to the randomized control trial design described in Chapter 2.4.

Quiz Chapter 5.1

Experiments

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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