Common misconceptions about data analysis and statistics
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Abstract
Ideally, any experienced investigator with the right tools should be able to reproduce a finding published in a peerreviewed biomedical science journal. In fact, the reproducibility of a large percentage of published findings has been questioned. Undoubtedly, there are many reasons for this, but one reason maybe that investigators fool themselves due to a poor understanding of statistical concepts. In particular, investigators often make these mistakes: 1. PHacking. This is when you reanalyze a data set in many different ways, or perhaps reanalyze with additional replicates, until you get the result you want. 2. Overemphasis on P values rather than on the actual size of the observed effect. 3. Overuse of statistical hypothesis testing, and being seduced by the word “significant”. 4. Overreliance on standard errors, which are often misunderstood.
Introduction
Ideally, any experienced investigator with the right tools should be able to reproduce a finding published in a peerreviewed biomedical science journal. In fact, the reproducibility of a large percentage of published findings has been questioned. Investigators at Bayer Healthcare were reportedly able to reproduce only 20–25 % of 67 preclinical studies (Prinz et al. 2011), and investigators at Amgen were able to reproduce only 6 of 53 studies in basic cancer biology despite often cooperating with the original investigators (Begley and Ellis 2012). This problem has been featured in a cover story in The Economist (Anonymous 2013) and has attracted the attention of the NIH leaders (Collins and Tabak 2014).
Why can so few findings be reproduced? Undoubtedly, there are many reasons. But in many cases, I suspect that investigators fooled themselves due to a poor understanding of statistical concepts (see Marino 2014, for a good review of this topic). Here, I identify five common misconceptions about statistics and data analysis, and explain how to avoid them. My recommendations are written for pharmacologists and other biologists publishing experimental research using commonly used statistical methods. They would need to be expanded for analyses of clinical or observational studies and for Bayesian analyses. This editorial is about analyzing and displaying data, so it does not address issues of experimental design.
My experience comes from basic pharmacology research conducted decades ago, followed by 25 years of answering email questions from scientists needing help analyzing data with GraphPad Prism,^{1} and authoring three editions of the text Intuitive Biostatistics (Motulsky 2014a).
Misconception 1: PHacking is OK
The results from data collected this way cannot be interpreted at face value. Even if there really is no difference (or no effect), the chance of finding a “statistically significant” result exceeds 5 %. The problem is that you introduce bias when you choose to collect more data (or analyze the data differently) only when the P value is greater than 0.05. If the P value was less than 0.05 in the first analyses, it might be larger than 0.05 after collecting more data or using an alternative analysis. But you would never see this if you only collected more data or tried different data analysis strategies when the first P value was greater than 0.05.
Exploring your data can be a very useful way to generate hypotheses and make preliminary conclusions. But all such analyses need to be clearly labeled, and then retested with new data.
 Ad hoc sample size selection. This is when you did not choose a sample size in advance, but just kept going until you liked the results. Figure 2 demonstrates the problem with ad hoc sample size determination. Distinguish unplanned ad hoc sample size decisions from planned “adaptive” sample size methods that make you “pay” for the increased versatility in sample size collection by requiring a stronger effect to reach “significance” (Kairalla et al. 2012; FDA 2010).
 Hypothesizing after the result is known (HARKing; Kerr 1998). This is when you analyze the data in many different ways (say different subgroups), discover an intriguing relationship, and then publish the data so it appears that the hypothesis was stated before the data were collected (Fig. 3). This is a form of multiple comparisons (Berry 2007). Kriegeskorte et al. (2009) call this double dipping, as you are using the same data both to generate a hypothesis and to test it.

Phacking. This is a general term that encompasses dynamic sample size collection, HARKing, and more. It was coined by Simmons et al. (2011) who also use the phrase, “too many investigator degrees of freedom.” Phacking is especially misleading when it involves changing the actual values analyzed. Examples include ad hoc sample size selection (see above), switching to an alternate control group (if you do not like the first results and your experiment involved two or more control groups), trying various combinations of independent variables to include in a multiple regression (whether the selection is manual or automatic), and analyzing various subgroups of the data. Reanalyzing a single data set in various ways is also Phacking but will not usually mislead you quite as much.

For each figure or table, clearly state whether or not the sample size was chosen in advance, and whether every step used to process and analyze the data was planned as part of the experimental protocol.

If you use any form of Phacking, label the conclusions as “preliminary.”
Misconception 2: P values convey information about effect size
The dependence of P values on sample size can lead to two problems.
A large P value is not proof of no (or little) effect
Identical P values with very different interpretations
Treatment 1 (mean ± SD, n)  Treatment 2 (mean ± SD, n)  Difference between means  P value  95 % CI of the difference between means  

Experiment A  1,000 ± 100, n = 50  990.0 ± 100, n = 50  10  0.6  −30 to 50 
Experiment B  1,000 ± 100, n = 3  950.0 ± 100, n = 3  50  0.6  −177 to 277 
Experiment C  100 ± 5.0, n = 135  102 ± 5.0, n = 135  2  0.001  0.8 to 3.2 
Experiment D  100 ± 5.0, n = 3  135 ± 5.0, n = 3  35  0.001  24 to 46 
In experiment A (from Table 1), the difference between means in the experimental sample is 10, so the difference equals 1 % of the mean of treatment 1. Assuming random sampling from Gaussian populations, the 95 % confidence interval for the difference between the two population means ranges from −30 to 50. In other words, the data are consistent (with 95 % confidence) with a decrease of 3 %, an increase of 5 %, or anything in between. The interpretation depends on the scientific context and the goals of the experiment, but in most contexts, these results can be summarized simply: The data are consistent with a tiny decrease, no change, or a tiny increase. These are solid negative data.
Experiment B is very different. The difference between means is larger, and the confidence interval is much wider (because the sample size is so small). Assuming random sampling from Gaussian populations, the data are consistent (with 95 % confidence) with anything between a decrease of 18 % and an increase of 28 %. The data are consistent with a large decrease, a small decrease, no difference, a small increase, or a large increase. These data lead to no useful conclusion at all! An experiment like this should not be published.
A small P value is not proof of a large effect
The bottom two rows of Table 1 presents the results of two simulated experiments where both P values are 0.001, but again two experiments lead to very different conclusions.
In experiment C (from Table 1), the difference between means in the experimental sample is only 2 (so the difference equals 2 % of the mean of treatment 1). Assuming random sampling from Gaussian populations, the 95 % confidence interval for the difference between the two population means ranges from 0.8 to 3.2. In other words, the data are consistent (with 95 % confidence) with anything between an increase of 0.8 % and an increase of 3.2 %. How to interpret that depends on the scientific context and the goals of the experiment, but in most contexts, this can be summarized simply: The data clearly demonstrate an increase, but that increase is tiny.
Experiment D is very different. The difference between means is 35 (so 35 % of the control mean), and the confidence interval extends from an increase of 23.7 % to an increase of 46.3 %. The data clearly demonstrate that there is an increase that is (with 95 % confidence) substantial.

Always show and emphasize the effect size (as difference, percent difference, ratio, or correlation coefficient) along with its confidence interval.

Consider omitting reporting of P values.
Misconception 3: statistical hypothesis testing and reports of “statistical significance” are necessary in experimental research
Statistical hypothesis testing is a way to make a crisp decision from one analysis. If the P value is less than a preset value (usually 0.05), the result is deemed “statistically significant” and you make one decision. Otherwise, the result is deemed “not statistically significant” and you make the other decision. This is helpful in quality control and some clinical studies. It also is useful when you rigorously compare the fits of two scientifically sensible models to your data, and choose one to guide your interpretation of the data and to plan future experiments.

The need to make a crisp decision based on one analysis is rare in basic research. A decision about whether or not to place an asterisk on a figure does not count! If you are not planning to make a crisp decision, the whole idea of statistical hypothesis testing is not helpful.

Statistical hypothesis testing “does not tell us what we want to know, and we so much want to know what we want to know that, out of desperation, we nevertheless believe that it does!” (Cohen 1994). Statistical hypothesis testing has even been called a cult (Ziliak and McCloskey 2008). The question we want to answer is: Given these data, how likely is the null hypothesis? The question that a P values answers is: Assuming the null hypothesis is true, how unlikely are these data? These two questions are distinct, and so have distinct answers.

Scientists who intend to use statistical hypothesis testing often end up not using it. If the P value is just a bit larger than 0.05, scientists often avoid the strict use of hypothesis testing and instead apply the “timehonored tactic of circumlocution to disguise the nonsignificant result as something more interesting” (Hankins 2013). They do this by using terms such as “almost significant,” “bordered on being statistically significant,” “a statistical trend toward significance,” or “approaching significance.” Hankins lists 468 such phrases he found in published papers!
 The 5 % significance threshold is often misunderstood. If you use a P value to make a decision, of course it is possible that you will make the wrong decision. In some cases, the P value will be tiny just by chance, even though the null hypothesis of no difference is actually true. In these cases, a conclusion that a finding is statistically significant is a false positive and you will have made what is called a type I error.^{2} Many scientists mistakenly believe that the chance of making a false positive conclusion is 5 %. In fact, in many situations, the chance of making a type I false positive conclusion is much higher than 5 % (Colquhoun 2014). For example, in a situation where you expect the null hypothesis to be true 90 % of the time (say you are screening lightly prescreened compounds, so expect 10 % to work), you have chosen a sample size large enough to ensure 80 % power, and you use the traditional 5 % significance level, the false discovery rate is not 5 % but rather is 36 % (the calculations are shown in Table 2). If you only look at experiments where the P value is just a tiny bit less than 0.050, the probability of a false positive rises to 79 % (Motulsky 2014b). Ioannidis (2005) used calculations like these (and other considerations) to argue that most published research findings are probably false.Table 2
The false discovery rate when P < 0.05
P < 0.05
P > 0.05
Total
Really is an effect
80
20
100
No effect (null hypothesis true)
45
855
900
Total
125
875
1,000

The word “significant” is often misunderstood. The problem is that “significant” has two distinct meanings in science (Motulsky 2014c). One meaning is that a P value is less than a preset threshold (usually 0.05). The other meaning of “significant” is that an effect is large enough to have a substantial physiological or clinical impact. These two meanings are completely different, but are often confused.

Only report statistical hypothesis testing (and place significance asterisks on figures) when you will make a decision based on that one analysis.

Never use the word “significant” in a scientific paper. If you use statistical hypothesis testing to make a decision, state the P value, your preset P value threshold, and your decision. When discussing the possible physiological or clinical impacts of a finding, use other words.
Misconception 4: the standard error of the mean quantifies variability
Pharmacology journals are full of graphs and tables showing the mean and the standard error of the mean (SEM).
A quick review. The standard deviation (SD) quantifies variation among a set of values, but the SEM does not. The SEM is computed by dividing the SD by the square root of sample size. With large samples, the SEM will be tiny even if there is a lot of variability.
One problem with plotting or displaying the mean ± SEM is that some people viewing the graph or table may mistakenly think that the error bars show the variability of the data. A second problem with reporting means with SEM is that the range mean ± SEM cannot be rigorously interpreted. The SEM gives information about how precisely you have determined the population mean. So the range mean ± SEM is a confidence interval, but the confidence level depends on sample size. With large samples, that range is a 68 % CI of the mean. When n = 3, that range is only a 58 % CI.^{3}

If you want to display the variability among the values, show raw data (which is not done often enough in my opinion). If showing the raw data would make the graph hard to read, show instead a boxwhisker plot, a frequency distribution, or the mean and SD.
 If you want readers to see how precisely you have determined the mean, report a confidence interval (95 % confidence intervals are standard). Figure 5 shows a dataset plotted using all of these methods.

When reporting results from regression, show the 95 % confidence interval of each parameter rather than standard errors.
Misconception 5: you do not need to report the details
The methods section of every paper should report the methods with enough detail that someone else could reproduce your work. This applies to statistical methods just as it does to experimental methods.

When reporting a sample size, explain exactly what you counted. Did you count replicates in one experiment (technical replicates), repeat experiments, the number of studies pooled in a metaanalysis, or something else?

If you eliminated any outliers, state how many outliers you eliminated, the rule used to identify them, and a statement whether this rule was chosen before collecting data.

If you normalized data, explain exactly how you defined 100 and 0 %.

When possible, report the P value up to at least a few digits of precision, rather than just stating whether the P value is less than or greater than an arbitrary threshold. For each P value, state the null hypothesis it tests if there is any possible ambiguity.

When reporting a P value that compares two groups, state whether the P value is one or twotailed. If you report a onetailed P value, state that you recorded a prediction for the direction of the effect (for example increase or decrease) before you collected any data and what this prediction was. If you did not record such a prediction, report a twotailed P value.

Explain the details of the statistical methods you used. For example, if you fit a curve using nonlinear regression, explain precisely which model you fit to the data and whether (and how) data were weighted. Also state the full version number and platform of the software you use.

Consider posting files containing both the raw data and the analyses so other investigators can see the details.
Summary
The physicist E. Rutherford supposedly said, “If your experiment needs statistics, you ought to have done a better experiment^{4}.” There is a lot of truth to that statement when you are working in a field with a very high signaltonoise ratio. In these fields, statistical analysis may not be necessary. But if you work in a field with a lower signaltonoise ratio, or are trying to compare the fits of alternative models that do not differ all that much, you need statistical analyses to properly quantify your confidence in your conclusions.
I suspect that one of the reasons that the results reported in many papers cannot be reproduced is that statistical analyses are often done as a quick afterthought, with the goal to sprinkle asterisks on figures and the word “significant” on conclusions. The suggestions I propose in this commentary can all be summarized simply: If you are going to analyze your data using statistical methods, then plan the methods carefully, do the analyses seriously, and report the data, methods, and results completely.
Footnotes
 1.
 2.
In contrast, a type II, or false negative, error is when there really is a difference but the result in your experiment is not statistically significant.
 3.
Computed using this Excel formula: = (1T.DIST.2T (1.0,2)). The first argument (1.0) is the number of SEMs (in each direction) included in the confidence interval, and the second argument (2) is the number of degrees of freedom, which equals n1.
 4.
The quotation is widely attributed to this famous physicist, but I cannot find an actual citation.
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