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Start by selecting a test appropriate for the experimental design under ideal circumstances. If the actual data collected do not meet these assumptions, one option is to change to an appropriate statistical test as discussed in Step 4 and illustrated in Example 1 of the Supplemental Tutorial.

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In addition to matching variables with types of tests, it is also important to make sure that the null and alternative hypotheses for a test will address your biological hypothesis. Most common statistical tests require predetermining an acceptable rate of false positives.

The type I error rate is adjusted to a lower value when multiple tests are being performed to address a common biological question Dudoit and van der Laan, Otherwise, lowering the type I error rate is not recommended, because it decreases the power of the test to detect small effects of treatments see below. Biological replicates measurements on separate samples are used for parameter estimates and statistical tests, because they allow one to describe variation in the population. Technical replicates multiple measurements on the same sample are used to improve estimation of the measurement for each biological replicate.

Treating technical replicates as biological replicates is called pseudoreplication and often produces low estimates of variance and erroneous test results. The difference between technical and biological replicates depends on how one defines the population of interest. For example, measurements on cells within one culture flask are considered to be technical replicates, and each culture flask to be a biological replicate, if the population is all cells of this type and variability between flasks is biologically important.

But in another study, cell to cell variability might be of primary interest, and measurements on separate cells within a flask could be considered biological replicates as long as one is cautious about making inferences beyond the population in that flask. Typically, one considers biological replicates to be the most independent samples. The design should be balanced in the sense of collecting equal numbers of replicates for each treatment.

Balanced designs are more robust to deviations from hypothesis test assumptions, such as equal variances in responses between treatments Table 1. Extensive replication of experiments large numbers of observations has bountiful virtues, including higher precision of parameter estimates, more power of statistical tests to detect small effects, and ability to verify the assumptions of statistical tests.

Fortunately, statistical analysis in experimental biology has two major advantages over observational biology.

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First, experimental conditions are often well controlled, for example using genetically identical organisms under laboratory conditions or administering a precise amount of a drug. This reduces the variation between samples and compensates to some extent for small sample sizes. Second, experimentalists can randomize the assignment of treatments to their specimens and therefore minimize the influence of confounding variables. Nonetheless, small numbers of observations make it difficult to verify important assumptions and can compromise the interpretation of an experiment.

One can estimate the appropriate number of measurements required by calculating statistical power when designing each experiment. Statistical power is the probability of rejecting a truly false null hypothesis. A common target is 0. Three variables contribute to statistical power: number of measurements, variability of those measurements SD , and effect size mean difference in response between the control and the treated populations.

A simple rule of thumb is that power decreases with the variability and increases with sample size and effect size as shown in Figure 4. One can increase the power of an experiment by reducing measurement error variance or increasing the sample size.

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For the statistical tests in Table 1 , simple formulas are available in most statistical software packages e. FIGURE 4: Three graphs show factors affecting the statistical power, the probability of rejecting a truly false null hypothesis in a two-sample t test. Two variables are held constant in each example.

Of course, one does not know the outcome of an experiment before it is done, but one may know the expected variability in the measurements from previous experiments, or one can run a pilot experiment on the control sample to estimate the variability in the measurements in a new system. Then one can design the experiment knowing roughly how many measurements will be required to detect a certain difference between the control and experimental samples. Alternatively, if the sample size is fixed, one can rearrange the power formula to compute the effect size one could detect at a given power and variability.

If this effect size is not meaningful, proceeding is not advised. Experimental data should not deviate strongly from the assumptions of the chosen statistical test Table 1 , and the sample sizes should be large enough to evaluate if this is the case. Strong deviations from expectations will result in inaccurate test results. Even a very well-designed experiment may require adjustments to the data analysis plan, if the data do not conform to expectations and assumptions.

See Examples 1, 2, and 4 in the Supplemental Tutorial. For example, a t test calls for continuous numerical data and assumes that the responses have a normal distribution Figure 1, A and B with equal variances for both treatments. Samples from a population are never precisely normally distributed and rarely have identical variances. How can one tell whether the data are meeting or failing to meet the assumptions?

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Find out whether the measurements are distributed normally by visualizing the unprocessed data. For numerical data this is best done by making a histogram with the range of values on the horizontal axis and frequency count of the value on the vertical-axis Figure 1B. Most statistical tests are robust to small deviations from a perfect bell-shaped curve, so a visual inspection of the histogram is sufficient, and formal tests of normality are usually unnecessary. The main problem encountered at this point in experimental biology is that the number of measurements is too small to determine whether they are distributed normally.

Not all data are distributed normally. A common deviation is a skewed distribution where the distribution of values around the peak value is asymmetrical Figure 1C. In many cases asymmetric distributions can be made symmetric by a transformation such as taking the log, square root, or reciprocal of the measurements for right-skewed data, and the exponential or square of the measurements for left-skewed data. For example, an experiment measuring cell division rates might result in many values symmetrically distributed around the mean rate but a long tail of much lower rates from cells that rarely or never divide.

A log transformation Figure 1D would bring the histogram of this data closer to a normal distribution and allow for more statistical tests. See Example 2 in the Supplemental Tutorial for an example of a log transformation. Exponential Figure 1E and bimodal Figure 1F distributions are also common. One can evaluate whether variances differ between treatments by visual inspection of histograms of the data or calculating the variance and SD for each treatment.

If the sample sizes are equal between treatments i.

To determine whether the assumption of linearity in regression has been met, one can look at a plot of residuals i. Residuals should be roughly uniform across fitted values, and deviations from uniform fitted values suggest nonlinearity.

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When nonlinearity is observed, one can consider more complicated parametric models of the relationship of responses and treatments. If the data do not meet the assumptions or sample sizes are too small to verify that assumptions have been met, alternative tests are available. If the responses are not normally distributed such as a bimodal distribution, Figure 1F , the Mann-Whitney U test can replace the t test, and the Kruskal-Wallis test can replace ANOVA with the assumption of consistently distributed responses across treatments.

However, relaxing the assumptions in such nonparametric tests reduces the power to detect the effects of treatments. See Supplemental Tutorial Example 1 for an example. Categorical tests typically only assume sample sizes are large enough to avoid low expected numbers of observations in each category. It is important to confirm that these assumptions have been met, so larger samples can be collected, if they have not been met.

A hypothesis test is done to determine the probability of observing the experimental data, if the null hypothesis is true. Such tests compare the properties of the experimental data with a theoretical distribution of outcomes expected when the null hypothesis is true. Note that different tests are required depending on whether the treatments and responses are categorical or numerical Table 1.

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## Introduction to Statistical Methods in AI

One example is the t test used for continuous numerical responses. In this case the properties of the data are summarized by a t statistic and compared with a t distribution Figure 5. The t distribution gives the probability of obtaining a given t statistic upon taking many random samples from a population where the null hypothesis is true.

The shape of the distribution depends on the sample sizes. The vertical dashed lines are 2. The t distribution is the theoretical probability of obtaining a given t statistic with many random samples from a population where the null hypothesis is true.

The shape of the distribution depends on the sample size. The distribution is symmetric, centered on 0. The tails are thicker than a standard normal distribution, reflecting the higher chance of values away from the mean when both the mean and the variance are being estimated from a sample. The t distribution is a probability density function so the total area under the curve is equal to 1.