Jul 9, 2014

Response to Jason Mitchell’s “On the Emptiness of Failed Replications”


Jason Mitchell recently wrote an article entitled “On the Emptiness of Failed Replications.”. In this article, Dr. Mitchell takes an unconventional and extremely strong stance against replication, arguing that: “… studies that produce null results -- including preregistered studies -- should not be published.” The crux of the argument seems to be that "scientists who get p > .05 are just incompetent." It completely ignores the possibility that a positive result could also (maybe even equally) be due to experimenter error. Dr. Mitchell also appears to ignore the possibility of simply getting a false positive (which is expected to happen under the null in 5% of cases).

More importantly, it ignores issues of effect size and treats the outcome of research as a dichotomous "success or fail.” The advantages of examining effect sizes over simple directional hypotheses using null hypothesis significance testing are beyond the scope of this short post, but you might check out Sullivan and Feinn (2012) as an open-access starting point. Generally speaking, the problem is that sampling variation means that some experiments will find null results even when the experimenter does everything right. As an illustration, below is 1000 simulated correlations, assuming that r = .30 in the population, and a sample size of 100 (I used a monte carlo method).

In this picture, the units of analysis are individual correlations obtained in 1 of 1000 hypothetical research studies. The x-axis is the value of the correlation coefficient found, and the y-axis is the number of studies reporting that value. The red line is the critical value for significant results at p < .05 assuming a sample size of 100. As you can see from this picture, the majority of studies are supportive of an effect that is greater than zero. However (simply due to chance) all the studies to the left of the red line turned out non-significant. If we suppressed all the null results (i.e., all those unlucky scientists to the left of the red line) as Dr. Mitchell suggests, then our estimate of the effect size in the population would be inaccurate; specifically, it would appear to be larger than it really is, because certain aspects of random variation (i.e., null results) are being selectively suppressed. Without the minority of null findings (in addition to the majority of positive findings) the overall estimate of the effect cannot be correctly estimated.

The situation is even more grim if there really is no effect in the population.

In this case, a small proportion of studies will produce false positives, with a roughly equal chance of an effect in either direction. If we fail to report null results, false positives may be reified as substantive effects. The reversal of signs across repeated studies might be a warning sign that the effect doesn’t really exist, but without replication, a single false positive could define a field if it happens (by chance) to be in line with prior theory.

With this in mind, I also disagree that replications are “publicly impugning the scientific integrity of their colleagues.” Some people feel threatened or attacked by replication. The ideas we produce as scientists are close to our hearts, and we tend to get defensive when they’re challenged. If we focus on effect sizes, rather than the “success or fail” logic of null hypothesis significance testing, then I don’t believe that “failed” replications damage the integrity of the original author, but rather simply suggests that we should modulate the estimate of the effect size downwards. In this framework, replication is less about “proving someone wrong” and more about centering on the magnitude of an effect size.

Something that is often missed in discussion of replication is that the very nature of randomness inherent in the statistical procedures scientists use means that any individual study (even if perfectly conducted) will probably generate an effect size that is a bit larger or smaller than it is in the population. It is only through repeated experiments that we are able to center on an accurate estimate of the effect size. This issue is independent of researcher competence, and means that even the most competent researchers will come to the wrong conclusions occasionally because of the statistical procedures and cutoffs we’ve chosen to rely on. With this in mind, people should be aware that a failed replication does not necessarily mean that one of the two researchers is incorrect or incompetent – instead, it is assumed (until further evidence is collected) that the best estimate is a weighted average of the effect size from each research study.

For some more commentary from other bloggers, you might check out the following links:

"Are replication efforts pointless?" by Richard Tomsett

"Being as wrong as can be on the so-called replication crisis of science" by drugmonkey at Scientopia

"Are replication efforts useless?" by Jan Moren

"Jason Mitchell’s essay" by Chris Said

"#MethodsWeDontReport – brief thought on Jason Mitchell versus the replicators" by Micah Allen

"On 'On the emptiness of failed replications'" by Neuroskeptic

Jan 29, 2014

Privacy in the Age of Open Data


Nothing is really private anymore. Corporations like Facebook and Google have been collecting our information for some time, and selling it in aggregate to the highest bidder. People have been raising concerns over these invasions of privacy, but generally only technically-savvy, highly motivated people can really be successful at remaining anonymous in this new digital world.

For a variety of incredibly important reasons, we are moving towards open research data as a scientific norm – that is, micro datasets and statistical syntax openly available to anyone who wants it. However, some people are uncomfortable with open research data, because they have concerns about privacy and confidentiality violations. Some of these violations are even making the news: A high profile case about people being identified from their publicly shared genetic information comes to mind.

With open data comes increased responsibility. As researchers, we need to take particular care to balance the advantages of data-sharing with the need to protect research participants from harm. I’m particularly primed for this issue because my own research often intersects with clinical psychology. I ask questions about things like depression, anxiety, eating disorders, substance use and conflict with romantic partners. The data collected in many of my studies has the potential to seriously harm the reputation – and potentially the mental health – of participants if linked to their identity by a malicious person. This said, I believe in the value of open data sharing. In this post, I’m going to discuss a few core issues as it pertains to de-identification – that is, ensuring the anonymity of participants in an openly shared dataset. Violations of privacy will always be a risk: However, some relatively simple steps on the part of the researcher can make re-identification of individual participants much more challenging.

Who are we protecting the data from?

Throughout the process, it’s helpful to imagine yourself as a person trying to get dirt on a potential participant. Of course, this is ignoring the fact that very few people are likely to use data for malicious purposes … but for now, let’s just consider the rare cases where this might happen. It only takes one high-profile incident to be a public relations and ethics nightmare for your research! There are two possibilities for malicious users that I can think of:

  1. Identity thieves who don’t know the participant directly, but are looking for enough personal information to duplicate someone’s identity for criminal activities, such as credit card fraud. These users are unlikely to know anything about participants ahead of time, so they have a much more challenging job because they have to be able to identify people exclusively using publicly available information.

  2. People who know the participant in real-life and want to find out private information about someone for some unpleasant purpose (e.g., stalkers, jealous romantic partners, a fired employee, etc.). In this case, the party likely knows (a) that the person of interest is in your dataset; (b) basic demographic information on the person such as sex, age, occupation, and the city they live in. Whether or not this user is successful in identifying individuals in an open dataset depends on what exactly the researcher has shared. For fine-grained data, it could be very easy; however, for properly de-identified data, it should be virtually impossible.

Key Identifiers to Consider when De-Identifying Data

The primary way to safeguard privacy in publicly shared data is to avoid identifiers; that is, pieces of information that can be used directly or indirectly to determine a person’s identity. A useful starting point for this is the list of 18 identifiers indicated in the Health Insurance Portability and Accountability Act that are to be used with Protected Health Information. A full list of these identifiers can be found here. Many of these identifiers are obvious (e.g., no names, phone numbers, SIN numbers, etc.), but some identifiers are worth discussing more specifically in the context of psychological research paradigm which shares data openly.

Demographic variables. Most of the variables that psychologists are interested in are not going to be very informative for identifying individuals. For example, reaction time data (even if unique to an individual) is very unlikely to identify participants – and in any event, most people are unlikely to care if other people know that they respond 50ms faster to certain types of visual stimuli. The type of data that are generally problematic are what I’ll call “demographic variables.” So things like sex, ethnicity, age, occupation, university major, etc. These data are sometimes used in analyses, but most often are just used to characterize the sample in the participants section of manuscripts. Most of the time, demographic variables can’t be used in isolation to identify people; instead, combinations of variables are used (e.g., a 27-year old, Mexican woman who works as a nurse may be the only person with that combination of traits in the data, leaving her vulnerable to loss of privacy). Because the combination of several demographic characteristics can potentially produce identifiable profiles, a common rule of thumb I picked up when working with Statistics Canada is to require a minimum of 5 participants per cell. In other words, if a particular combination of demographic features yields less than 5 individuals, the group will be collapsed into a larger, more anonymous, aggregate group. The most common example of this would be using age ranges (e.g., ages 18-25) instead of exact ages; similar logic could apply to most demographic variables. This rule can get restrictive fast (but also demonstrates how little data can be required to identify individual people!) so ideally, share only the demographic information that is theoretically and empirically important to your research area.

Outliers and rare values. Another major issue are outliers and other rare values. Outliers are variably defined depending on the statistical text you read, but generally refer to extreme values when variables are using continuous, interval, or ordinal measurement (e.g., someone has an IQ of 150 in your sample, and the next highest person is 120). Rare values refer to categorical data that very few people endorse (e.g., the only physics professor in a sample). There are lots of different ways you can deal with outliers, and there’s not necessarily a lot of agreement on which is the best – indeed, it’s one of those researcher degrees of freedom you might have heard about. Though this may depend on the sensitivity of the data in question, outliers often have the potential to be a privacy risk. From a privacy standpoint, it may be best for the researcher to deal with outliers by deleting or transforming them before sharing the data. For rare values, you can collapse response options together until there are no more unique values (e.g., perhaps classify the physics professor as a “teaching professional” if there are other teachers in the sample). In the worst case scenario, you may need to report the value as missing data (e.g., a single intersex person in your sample that doesn’t identify as male or female). Whatever you decide, you should disclose to readers what your strategy was for dealing with outliers and rare values in the accompanying documentation so it is clear for everyone using the data.

Dates. Though it might not be immediately obvious, any exact dates in the dataset place participants at risk for re-identification. For example, if someone knew what day the participant took part in a study (e.g., they mention it to a friend; they’re seen in a participant waiting area) then their data would be easily identifiable by this date. To minimize privacy risks, no exact dates should be included in the shared dataset. If dates are necessary for certain analyses, transforming the data into some less identifiable format that is still useful for analyses is preferable (e.g., have variables for “day of week” or “number of days in between measurement occasions” if these are important).

Geographic Locations. The rule of having “no geographic subdivisions smaller than a state” from the HIPAA guidelines is immediately problematic for many studies. Most researchers collect data from their surrounding community. Thus, it will be impossible to blind the geographic location in many circumstances (e.g., if I recruit psychology students for my study, it will be easy for others to infer that I did so from my place of employment at Dalhousie University). So at a minimum, people will know that participants are probably living relatively close to my place of employment. This is going to be unavoidable in many circumstances, but in most cases it should not be enough to identify participants. However, you will need to consider if this geographical information can be combined with other demographic information to potentially identify people, since it will not be possible to suppress this information in many cases. Aside from that, you’ll just have to do your best to avoid more finely grained geographical information. For example, in Canada, a reverse lookup of postal codes can identify some locations with a surprising degree of accuracy, sometimes down to a particular street!

Participant ID numbers. Almost every dataset will (and should) have a unique identification number for each participant. If this is just a randomly selected number, there are no major issues. However, most researchers I know generate ID numbers in non-random ways. For example, in my own research on romantic couples we assign ID numbers chronologically, with a suffix number of “1” indicating men and “2” indicating women. So ID 003-2 would be the third couple that participated, and the male within that couple. In this kind of research, the most likely person to snoop would probably be the other romantic partner. If I were to leave the ID numbers as originally entered, the romantic partner would easily be able to find their own partner’s data (assuming a heterosexual relationship and that participants remember their own ID number). There are many other algorithms researchers might use to create ID numbers, many of which do not provide helpful information to other researchers, but could be used to identify people. Before freely sharing data, you might consider scrambling the unique ID numbers so that they cannot be a privacy risk (you can, of course, keep a record of the original ID numbers in your own files if needed for administrative purposes).

Some Final Thoughts

Risk of re-identification is never zero. Especially when data are shared openly online, there will always be a risk for participants. Making sure participants are fully informed about the risks involved during the consent process is essential. Careless sharing of data could result in a breach of privacy, which could have extremely negative consequences both for the participants and for your own research program. However, with proper safeguards, the risk of re-identification is low, in part due to some naturally occurring features of research. The slow, plodding pace of scientific research inadvertently protects the privacy of participants: Databases are likely to be 1-3 years old by the time they are posted, and people can change considerably within that time, making them harder to identify. Naturally occurring noise (e.g., missing data, imputation, errors by participants) also impedes the ability to identify people, and the variables psychologists are usually most interested in are often not likely candidates to re-identify someone.

As a community of scientists devoted to making science more transparent and open, we also carry the responsibility of protecting the privacy and rights of participants as much as is possible. I don’t think we have all the answers yet, and there’s a lot more to consider when moving forward. Ethical principles are not static; there are no single “right” answers that will be appropriate for all research, and standards will change as technology and social mores change with each generation. Still, by moving forward with an open mind, and a strong ethical conscience to protect the privacy of participants, I believe that data can really be both open and private.

Dec 18, 2013

Researcher Degrees of Freedom in Data Analysis


The enormous amount of options available for modern data analysis is both a blessing and a curse. On one hand, researchers have specialized tools for any number of complex questions. On the other hand, we’re also faced with a staggering number of equally-viable choices, many times without any clear-cut guidelines for deciding between them. For instance, I just popped open SPSS statistical software and counted 18 different ways to conduct post-hoc tests for a one-way ANOVA. Some choices are clearly inferior (e.g., the LSD test doesn’t adjust p-values for multiple comparisons) but it’s possible to defend the use of many of the available options. These ambiguous choice points are sometimes referred to as researcher degrees of freedom.

In theory, researcher degrees of freedom shouldn’t be a problem. More choice is better, right? The problem arises from two interconnected issues: (a) Ambiguity as to which statistical test is most appropriate and (b) an incentive system where scientists are rewarded with publications, grants, and career stability when their p-values fall below the revered p < .05 criterion. So, perhaps unsurprisingly, when faced with a host of ambiguous options for data analysis, most people settle on the one that achieves statistically significant results. Simmons, Nelson, and Simonsohn (2011) argue that this undisclosed flexibility in data analysis allows people to present almost any data as “significant,” and calls for 10 simple guidelines for reviewers and authors to disclose in every paper – which, if you haven’t read yet are worth checking out. In this post, I will discuss a few guidelines of my own for conducting data analysis in a way that strives to overcome our inherent tendency to be self-serving.

  1. Make as many data analytic decisions as possible before looking at your data. Review the statistical literature and decide on which statistical test(s) will be best before looking at your collected data. Continue to use those tests until enough evidence emerges to change your mind. The important thing is that you make these decisions before looking at your data. Once you start playing with the actual data, your self-serving biases will start to kick in. Do not underestimate your ability for self-deception: Self-serving biases are powerful, pervasive, and apply to virtually everyone. Consider pre-registering your data analysis plan (perhaps using the Open Science Framework to keep yourself honest and to convince future reviewers that you aren’t exploiting researcher degrees of freedom.

  2. When faced with a situation where there are too many equally viable choices, run a small number of the best choices, and report all of them. In this case, decide on 2-5 different tests ahead of time. Report the results of all choices, and make a tentative conclusion based if the majority of these tests agree. For instance, when determining model fit in structural equation modeling, there many different methods you might use. If you can’t figure out which method is best by reviewing the statistical literature – it’s not entirely clear, statisticians disagree about as often as any other group of scientists – then report the results of all tests, and make a conclusion if they all converge on the same solution. When they disagree, make a tentative conclusion based on the majority of tests that agree (e.g., 2 of 3 tests come to the same conclusion). For the record, I currently use CFI, TLI, RMSEA, and SRMR in my own work, and use these even if other fit indices provide more favorable results.

  3. When deciding on a data analysis plan after you’ve seen the data, keep in mind that most researcher degrees of freedom have minimal impact on strong results. For any number of reasons, you might find yourself deciding on a data analysis plan after you’ve played around with the data for a while. At the end of the day, strong data will not be influenced much by researcher degrees of freedom. For instance, results should look much the same regardless of whether you exclude outliers, transform them, or leave them in the data when you have a study with high statistical power. Simmons et al. (2011) specifically recommend that results should be presented (a) with and without covariates, and (b) with and without specific data points excluded, if any were removed. Again, the general idea is that strong results will not change much when you alter researcher degrees of freedom. Thus, I again recommend analyzing the data in a few different ways and looking for convergence across all methods when you’re developing a data analysis plan after seeing the data. This sets the bar higher to try and combat your natural tendency to report just the one analysis that “works.” When minor data analytic choices drastically change the conclusions, this should be a warning sign that your solution is unstable and the results are probably not trustworthy. The number one reason why you have an unstable solution is probably because you have low statistical power. Since you hopefully had a strict data collection end date, the only viable alternative when results are unstable is to replicate the results in a second, more highly-powered study using the same data analytic approach.

At the end of the day, there is no “quick-fix” for the problem of self-serving biases during data analysis so long as the incentive system continues to reward novel, statistically significant results. However, by using the tips in this article (and elsewhere) researchers can focus on finding strong, replicable results by minimizing the natural human tendency to be self-serving.


Simmons, J. P., Nelson, L. D., & Simonsohn, U. (2011). False-positive psychology: Undisclosed flexibility in data collection and analysis allows presenting anything as significant. Psychological Science, 22, 1359-1366. doi:10.1177/0956797611417632

Nov 3, 2013

Increasing statistical power in psychological research without increasing sample size


What is statistical power and precision?

This post is going to give you some practical tips to increase statistical power in your research. Before going there though, let’s make sure everyone is on the same page by starting with some definitions.

Statistical power is the probability that the test will reject the null hypothesis when the null hypothesis is false. Many authors suggest a statistical power rate of at least .80, which corresponds to an 80% probability of not committing a Type II error.

Precision refers to the width of the confidence interval for an effect size. The smaller this width, the more precise your results are. For 80% power, the confidence interval width will be roughly plus or minus 70% of the population effect size (Goodman & Berlin, 1994). Studies that have low precision have a greater probability of both Type I and Type II errors (Button et al., 2013).

To get an idea of how this works, here are a few examples of the sample size required to achieve .80 power for small, medium, and large (Cohen, 1992) correlations as well as the expected confidence intervals

Population Effect Size Sample Size for 80% Power Estimated Precision
r = .10 782 95% CI [.03, .17]
r = .30 84 95% CI [.09, .51]
r = .50 29 95% CI [.15, .85]

Studies in psychology are grossly underpowered

Okay, so now you know what power is. But why should you care? Fifty years ago, Cohen (1962) estimated the statistical power to detect a medium effect size in abnormal psychology was about .48. That’s a false negative rate of 52%, which is no better than a coin-flip! The situation has improved slightly, but it’s still a serious problem today. For instance, one review suggested only 52% of articles in the applied psychology literature achieved .80 power for a medium effect size (Mone et al., 1996). This is in part because psychologists are studying small effects. One massive review of 322 meta-analyses including 8 million participants (Richard et al., 2003) suggested that the average effect size in social psychology is relatively small (r = .21). To put this into perspective, you’d need 175 participants to have .80 power for a simple correlation between two variables at this effect size. This gets even worse when we’re studying interaction effects. One review suggests that the average effect size for interaction effects is even smaller (f2 = .009), which means that sample sizes of around 875 people would be needed to achieve .80 power (Aguinis et al., 2005). Odds are, if you took the time to design a research study and collect data, you want to find a relationship if one really exists. You don’t want to "miss" something that is really there. More than this, you probably want to have a reasonably precise estimate of the effect size (it’s not that impressive to just say a relationship is positive and probably non-zero). Below, I discuss concrete strategies for improving power and precision.

What can we do to increase power?

It is well-known that increasing sample size increases statistical power and precision. Increasing the population effect size increases statistical power, but has no effect on precision (Maxwell et al., 2008). Increasing sample size improves power and precision by reducing standard error of the effect size. Take a look at this formula for the confidence interval of a linear regression coefficient (McClelland, 2000):

Power Equation

MSE is the mean square error, n is the sample size, Vx is the variance of X, and (1-R2) is the proportion of the variance in X not shared by any other variables in the model. Okay, hopefully you didn’t nod off there. There’s a reason I’m showing you this formula. In this formula, decreasing any value in the numerator (MSE) or increasing anything in the denominator (n, Vx, 1-R2) will decrease the standard error of the effect size, and will thus increase power and precision. This formula demonstrates that there are at least three other ways to increase statistical power aside from sample size: (a) Decreasing the mean square error; (b) increasing the variance of x; and (c) increasing the proportion of the variance in X not shared by any other predictors in the model. Below, I’ll give you a few ways to do just that.

Recommendation 1: Decrease the mean square error

Referring to the formula above, you can see that decreasing the mean square error will have about the same impact as increasing sample size. Okay. You’ve probably heard the term "mean square error" before, but the definition might be kind of fuzzy. Basically, your model makes a prediction for what the outcome variable (Y) should be, given certain values of the predictor (X). Naturally, it’s not a perfect prediction because you have measurement error, and because there are other important variables you probably didn’t measure. The mean square error is the difference between what your model predicts, and what the true values of the data actually are. So, anything that improves the quality of your measurement or accounts for potential confounding variables will reduce the mean square error, and thus improve statistical power. Let’s make this concrete. Here are three specific techniques you can use:

a) Reduce measurement error by using more reliable measures(i.e., better internal consistency, test-retest reliability, inter-rater reliability, etc.). You’ve probably read that .70 is the "rule-of-thumb" for acceptable reliability. Okay, sure. That’s publishable. But consider this: Let’s say you want to test a correlation coefficient. Assuming both measures have a reliability of .70, your observed correlation will be about 1.43 times smaller than the true population parameter (I got this using Spearman’s correlation attenuation formula). Because you have a smaller observed effect size, you end up with less statistical power. Why do this to yourself? Reduce measurement error. If you’re an experimentalist, make sure you execute your experimental manipulations exactly the same way each time, preferably by automating them. Slight variations in the manipulation (e.g., different locations, slight variations in timing) might reduce the reliability of the manipulation, and thus reduce power.

b) Control for confounding variables. With correlational research, this means including control variables that predict the outcome variable, but are relatively uncorrelated with other predictor variables. In experimental designs, this means taking great care to control for as many possible confounds as possible. In both cases, this reduces the mean square error and improves the overall predictive power of the model – and thus, improves statistical power. Be careful when adding control variables into your models though: There are diminishing returns for adding covariates. Adding a couple of good covariates is bound to improve your model, but you always have to balance predictive power against model complexity. Adding a large number of predictors can sometimes lead to overfitting (i.e., the model is just describing noise or random error) when there are too many predictors in the model relative to the sample size. So, controlling for a couple of good covariates is generally a good idea, but too many covariates will probably make your model worse, not better, especially if the sample is small.

c) Use repeated-measures designs. Repeated measures designs are where participants are measured multiple times (e.g., once a day surveys, multiple trials in an experiment, etc.). Repeated measures designs reduce the mean square error by partitioning out the variance due to individual participants. Depending on the kind of analysis you do, it can also increase the degrees of freedom for the analysis substantially. For example, you might only have 100 participants, but if you measured them once a day for 21 days, you’ll actually have 2100 data points to analyze. The data analysis can get tricky and the interpretation of the data may change, but many multilevel and structural equation models can take advantage of these designs by examining each measurement occasion (i.e., each day, each trial, etc.) as the unit of interest, instead of each individual participant. Increasing the degrees of freedom is much like increasing the sample size in terms of increasing statistical power. I’m a big fan of repeated measures designs, because they allow researchers to collect a lot of data from fewer participants.

Recommendation 2: Increase the variance of your predictor variable

Another less-known way to increase statistical power and precision is to increase the variance of your predictor variables (X). The formula listed above shows that doubling the variance of X is has the same impact on increasing statistical precision as doubling the sample size does! So it’s worth figuring this out.

a) In correlational research, use more comprehensive continuous measures. That is, there should be a large possible range of values endorsed by participants. However, the measure should also capture many different aspects of the construct of interest; artificially increasing the range of X by adding redundant items (i.e., simply re-phrasing existing items to ask the same question) will actually hurt the validity of the analysis. Also, avoid dichotomizing your measures (e.g., median splits), because this reduces the variance and typically reduces power (MacCallum et al., 2002).

b) In experimental research, unequally allocating participants to each condition can improve statistical power. For example, if you were designing an experiment with 3 conditions (let’s say low, medium, or high self-esteem). Most of us would equally assign participants to all three groups, right? Well, as it turns out, assigning participants equally across groups usually reduces statistical power. The idea behind assigning participants unequally to conditions is to maximize the variance of X for the particular kind of relationship under study -- which, according the formula I gave earlier, will increase power and precision. For example, the optimal design for a linear relationship would be 50% low, 50% high, and omit the medium condition. The optimal design for a quadratic relationship would be 25% low, 50% medium, and 25% high. The proportions vary widely depending on the design and the kind of relationship you expect, but I recommend you check out McClelland (1997) to get more information on efficient experimental designs. You might be surprised.

Recommendation 3: Make sure predictor variables are uncorrelated with each other

A final way to increase statistical power is to increase the proportion of the variance in X not shared with other variables in the model. When predictor variables are correlated with each other, this is known as colinearity. For example, depression and anxiety are positively correlated with each other; including both as simultaneous predictors (say, in multiple regression) means that statistical power will be reduced, especially if one of the two variables actually doesn’t predict the outcome variable. Lots of textbooks suggest that we should only be worried about this when colinearity is extremely high (e.g., correlations around > .70). However, studies have shown that even modest intercorrlations among predictor variables will reduce statistical power (Mason et al., 1991). Bottom line: If you can design a model where your predictor variables are relatively uncorrelated with each other, you can improve statistical power.


Increasing statistical power is one of the rare times where what is good for science, and what is good for your career actually coincides. It increases the accuracy and replicability of results, so it’s good for science. It also increases your likelihood of finding a statistically significant result (assuming the effect actually exists), making it more likely to get something published. You don’t need to torture your data with obsessive re-analysis until you get p < .05. Instead, put more thought into research design in order to maximize statistical power. Everyone wins, and you can use that time you used to spend sweating over p-values to do something more productive. Like volunteering with the Open Science Collaboration.


Aguinis, H., Beaty, J. C., Boik, R. J., & Pierce, C. A. (2005). Effect Size and Power in Assessing Moderating Effects of Categorical Variables Using Multiple Regression: A 30-Year Review. Journal of Applied Psychology, 90, 94-107. doi:10.1037/0021-9010.90.1.94

Button, K. S., Ioannidis, J. A., Mokrysz, C., Nosek, B. A., Flint, J., Robinson, E. J., & Munafò, M. R. (2013). Power failure: Why small sample size undermines the reliability of neuroscience. Nature Reviews Neuroscience, 14(5), 365-376. doi: 10.1038/nrn3475

Cohen, J. (1962). The statistical power of abnormal-social psychological research: A review. The Journal of Abnormal and Social Psychology, 65, 145-153. doi:10.1037/h0045186

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Goodman, S. N., & Berlin, J. A. (1994). The use of predicted confidence intervals when planning experiments and the misuse of power when interpreting results. Annals of Internal Medicine, 121, 200-206.

Hansen, W. B., & Collins, L. M. (1994). Seven ways to increase power without increasing N. In L. M. Collins & L. A. Seitz (Eds.), Advances in data analysis for prevention intervention research (NIDA Research Monograph 142, NIH Publication No. 94-3599, pp. 184–195). Rockville, MD: National Institutes of Health.

MacCallum, R. C., Zhang, S., Preacher, K. J., & Rucker, D. D. (2002). On the practice of dichotomization of quantitative variables. Psychological Methods, 7, 19-40. doi:10.1037/1082-989X.7.1.19

Mason, C. H., & Perreault, W. D. (1991). Collinearity, power, and interpretation of multiple regression analysis. Journal of Marketing Research, 28, 268-280. doi:10.2307/3172863

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McClelland, G. H. (1997). Optimal design in psychological research. Psychological Methods, 2, 3-19. doi:10.1037/1082-989X.2.1.3

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Richard, F. D., Bond, C. r., & Stokes-Zoota, J. J. (2003). One Hundred Years of Social Psychology Quantitatively Described. Review of General Psychology, 7, 331-363. doi:10.1037/1089-2680.7.4.331