Imagine an NBA player has an 80% chance to make one free throw. What is the chance that he makes both free throws? The correct answer is 64% (80% * 80%).
Now consider the possibility that it is possible to distinguish between two types of free throws. Some free throws are good; they don’t touch the rim and make a swishing sound when they go through the net (all net). The other free throws bounce of the rim and go in (rattling in).
What is the probability that an NBA player with an 80% free throw percentage makes a free throw that is all net or rattles in? It is more likely that an NBA player with an 80% free throw average makes a perfect free throw because a free throw that rattles in could easily have bounded the wrong way, which would lower the free throw percentage. To achieve an 80% free throw percentage, most free throws have to be close to perfect.
Let’s say the probability of hitting the rim and going in is 30%. With an 80% free throw average, this means that the majority of free throws are in the close-to-perfect category (20% misses, 30% rattle-in, 50% close-to-perfect).
What does this have to do with science? A lot!
The reason is that the outcome of a scientific study is a bit like throwing free throws. One factor that contributes to a successful study is skill (making correct predictions, avoiding experimenter errors, and conducting studies with high statistical power). However, another factor is random (a lucky or unlucky bounce).
The concept of statistical power is similar to an NBA players’ free throw percentage. A researcher who conducts studies with 80% statistical power is going to have an 80% success rate (that is, if all predictions are correct). In the remaining 20% of studies, a study will not produce a statistically significant result, which is equivalent to missing a free throw and not getting a point.
Many years ago, Jacob Cohen observed that researchers often conduct studies with relatively low power to produce a statistically significant result. Let’s just assume right now that a researcher conducts studies with 60% power. This means, researchers would be like NBA players with a 60% free-throw average.
Now imagine that researchers have to demonstrate an effect not only once, but also a second time in an exact replication study. That is researchers have to make two free throws in a row. With 60% power, the probability to get two significant results in a row is only 36% (60% * 60%). Moreover, many of the freethrows that are made rattle in rather than being all net. The percentages are about 40% misses, 30% rattling in and 30% all net.
One major difference between NBA players and scientists is that NBA players have to demonstrate their abilities in front of large crowds and TV cameras, whereas scientists conduct their studies in private.
Imagine an NBA player could just go into a private room, throw two free throws and then report back how many free throws he made and the outcome of these free throws determine who wins game 7 in the playoff finals. Would you trust the player to tell the truth?
If you would not trust the NBA player, why would you trust scientists to report failed studies? You should not.
It can be demonstrated statistically that scientists are reporting more successes than the power of their studies would justify (Sterling et al., 1995; Schimmack, 2012). Amongst scientists this fact is well known, but the general public may not fully appreciate the fact that a pair of exact replication studies with significant results is often just a selection of studies that included failed studies that were not reported.
Fortunately, it is possible to use statistics to examine whether the results of a pair of studies are likely to be honest or whether failed studies were excluded. The reason is that an amateur is not only more likely to miss a free throw. An amateur is also less likely to make a perfect free throw.
Based on the theory of statistical power developed by Nyman and Pearson and popularized by Jacob Cohen, it is possible to make predictions about the relative frequency of p-values in the non-significant (failure), just significant (rattling in), and highly significant (all net) ranges.
As for made-free-throws, the distinction between lucky and clear successes is somewhat arbitrary because power is continuous. A study with a p-value of .0499 is very lucky because p = .501 would have been not significant (rattled in after three bounces on the rim). A study with p = .000001 is a clear success. Lower p-values are better, but where to draw the line?
As it turns out, Jacob Cohen’s recommendation to conduct studies with 80% power provides a useful criterion to distinguish lucky outcomes and clear successes.
Imagine a scientist conducts studies with 80% power. The distribution of observed test-statistics (e.g. z-scores) shows that this researcher has a 20% chance to get a non-significant result, a 30% chance to get a lucky significant result (p-value between .050 and .005), and a 50% chance to get a clear significant result (p < .005). If the 20% failed studies are hidden, the percentage of results that rattled in versus studies with all-net results are 37 vs. 63%. However, if true power is just 20% (an amateur), 80% of studies fail, 15% rattle in, and 5% are clear successes. If the 80% failed studies are hidden, only 25% of the successful studies are all-net and 75% rattle in.
One problem with using this test to draw conclusions about the outcome of a pair of exact replication studies is that true power is unknown. To avoid this problem, it is possible to compute the maximum probability of a rattling-in result. As it turns out, the optimal true power to maximize the percentage of lucky outcomes is 66% power. With true power of 66%, one would expect 34% misses (p > .05), 32% lucky successes (.050 < p < .005), and 34% clear successes (p < .005).
For a pair of exact replication studies, this means that there is only a 10% chance (32% * 32%) to get two rattle-in successes in a row. In contrast, there is a 90% chance that misses were not reported or that an honest report of successful studies would have produced at least one all-net result (z > 2.8, p < .005).
Example: Unconscious Priming Influences Behavior
I used this test to examine a famous and controversial set of exact replication studies. In Bargh, Chen, and Burrows (1996), Dr. Bargh reported two exact replication studies (studies 2a and 2b) that showed an effect of a subtle priming manipulation on behavior. Undergraduate students were primed with words that are stereotypically associated with old age. The researchers then measured the walking speed of primed participants (n = 15) and participants in a control group (n = 15).
The two studies were not only exact replications of each other; they also produced very similar results. Most readers probably expected this outcome because similar studies should produce similar results, but this false belief ignores the influence of random factors that are not under the control of a researcher. We do not expect lotto winners to win the lottery again because it is an entirely random and unlikely event. Experiments are different because there could be a systematic effect that makes a replication more likely, but in studies with low power results should not replicate exactly because random sampling error influences results.
Study 1: t(28) = 2.86, p = .008 (two-tailed), z = 2.66, observed power = 76%
Study 2: t(28) = 2.16, p = .039 (two-tailed), z = 2.06, observed power = 54%
The median power of these two studies is 65%. However, even if median power were lower or higher, the maximum probability of obtaining two p-values in the range between .050 and .005 remains just 10%.
Although this study has been cited over 1,000 times, replication studies are rare.
One of the few published replication studies was reported by Cesario, Plaks, and Higgins (2006). Naïve readers might take the significant results in this replication study as evidence that the effect is real. However, this study produced yet another lucky success.
Study 3: t(62) = 2.41, p = .019, z = 2.35, observed power = 65%.
The chances of obtaining three lucky successes in a row is only 3% (32% *32% * 32*). Moreover, with a median power of 65% and a reported success rate of 100%, the success rate is inflated by 35%. This suggests that the true power of the reported studies is considerably lower than the observed power of 65% and that observed power is inflated because failed studies were not reported.
The R-Index corrects for inflation by subtracting the inflation rate from observed power (65% – 35%). This means the R-Index for this set of published studies is 30%.
This R-Index can be compared to several benchmarks.
An R-Index of 22% is consistent with the null-hypothesis being true and failed attempts are not reported.
An R-Index of 40% is consistent with 30% true power and all failed attempts are not reported.
It is therefore not surprising that other researchers were not able to replicate Bargh’s original results, even though they increased statistical power by using larger samples (Pashler et al. 2011, Doyen et al., 2011).
In conclusion, it is unlikely that Dr. Bargh’s original results were the only studies that they conducted. In an interview, Dr. Bargh revealed that the studies were conducted in 1990 and 1991 and that they conducted additional studies until the publication of the two studies in 1996. Dr. Bargh did not reveal how many studies they conducted over the span of 5 years and how many of these studies failed to produce significant evidence of priming. If Dr. Bargh himself conducted studies that failed, it would not be surprising that others also failed to replicate the published results. However, in a personal email, Dr. Bargh assured me that “we did not as skeptics might presume run many studies and only reported the significant ones. We ran it once, and then ran it again (exact replication) in order to make sure it was a real effect.” With a 10% probability, it is possible that Dr. Bargh was indeed lucky to get two rattling-in findings in a row. However, his aim to demonstrate the robustness of an effect by trying to show it again in a second small study is misguided. The reason is that it is highly likely that the effect will not replicate or that the first study was already a lucky finding after some failed pilot studies. Underpowered studies cannot provide strong evidence for the presence of an effect and conducting multiple underpowered studies reduces the credibility of successes because the probability of this outcome to occur even when an effect is present decreases with each study (Schimmack, 2012). Moreover, even if Bargh was lucky to get two rattling-in results in a row, others will not be so lucky and it is likely that many other researchers tried to replicate this sensational finding, but failed to do so. Thus, publishing lucky results hurts science nearly as much as the failure to report failed studies by the original author.
Dr. Bargh also failed to realize how lucky he was to obtain his results, in his response to a published failed-replication study by Doyen. Rather than acknowledging that failures of replication are to be expected, Dr. Bargh criticized the replication study on methodological grounds. There would be a simple solution to test Dr. Bargh’s hypothesis that he is a better researcher and that his results are replicable when the study is properly conducted. He should demonstrate that he can replicate the result himself.
In an interview, Tom Bartlett asked Dr. Bargh why he didn’t conduct another replication study to demonstrate that the effect is real. Dr. Bargh’s response was that “he is aware that some critics believe he’s been pulling tricks, that he has a “special touch” when it comes to priming, a comment that sounds like a compliment but isn’t. “I don’t think anyone would believe me,” he says.” The problem for Dr. Bargh is that there is no reason to believe his original results, either. Two rattling-in results alone do not constitute evidence for an effect, especially when this result could not be replicated in an independent study. NBA players have to make free-throws in front of a large audience for a free-throw to count. If Dr. Bargh wants his findings to count, he should demonstrate his famous effect in an open replication study. To avoid embarrassment, it would be necessary to increase the power of the replication study because it is highly unlikely that even Dr. Bargh can continuously produce significant results with samples of N = 30 participants. Even if the effect is real, sampling error is simply too large to demonstrate the effect consistently. Knowledge about statistical power is power. Knowledge about post-hoc power can be used to detect incredible results. Knowledge about a priori power can be used to produce credible results.
Updated on May 19, 2016
– corrected mistake in calculation of p-value for TIVA
A Replicability Analysis of Spencer, Steele, and Quinn’s seminal article on stereotype threat effects on gender differences in math performance.
In a seminal article, Spencer, Steele, and Quinn (1999) proposed the concept of stereotype threat. They argued that women may experience stereotype-threat during math tests and that stereotype threat can interfere with their performance on math tests.
The original study reported three experiments.
Study 1 had 56 participants (28 male and 28 female undergraduate students). The main aim was to demonstrate that stereotype-threat influences performance on difficult, but not on easy math problems.
A 2 x 2 mixed model ANOVA with sex and difficulty produced the following results.
Main effect for sex, F(1, 52) = 3.99, p = .051 (reported as p = .05), z = 1.96, observed power = 50%.
Interaction between sex and difficulty, F(1, 52) = 5.34 , p = .025, z = 2.24, observed power = 61%.
The low observed power suggests that sampling error contributed to the significant results. Assuming observed power is a reliable estimate of true power, the chance of obtaining significant results in both studies would only be 31%. Moreover, if the true power is in the range between 50% and 80% power, there is only a 32% chance that observed power to fall into this range. The chance that both observed power values fall into this range is only 10%.
Median observed power is 56%. The success rate is 100%. Thus, the success rate is inflated by 44 percentage points (100% – 56%).
The R-Index for these two results is low, Ř = 12 (56 – 44).
Empirical evidence shows that studies with low R-Indices often fail to replicate in exact replication studies.
It is even more problematic that Study 1 was supposed to demonstrate just the basic phenomenon that women perform worse on math problems than men and that the following studies were designed to move this pre-existing gender difference around with an experimental manipulation. If the actual phenomenon is in doubt, it is unlikely that experimental manipulations of the phenomenon will be successful.
The main purpose of Study 2 was to demonstrate that gender differences in math performance would disappear when the test is described as gender neutral.
Study 2 recruited 54 students (30 women, 24 men). This small sample size is problematic for several reasons. Power analysis of Study 1 suggested that the authors were lucky to obtain significant results. If power is 50%, there is a 50% chance that an exact replication study with the same sample size will produce a non-significant result. Another problem is that sample sizes need to increase to demonstrate that the gender difference in math performance can be influenced experimentally.
The data were not analyzed according to this research plan because the second test was so difficult that nobody was able to solve these math problems. However, rather than repeating the experiment with a better selection of math problems, the results for the first math test were reported.
As there was no repeated performance by the two participants, this is a 2 x 2 between-subject design that crosses sex and treat-manipulation. With a total sample size of 54 students, the n per cell is 13.
The main effect for sex was significant, F(1, 50) = 5.66, p = .021, z = 2.30, observed power = 63%.
The interaction was also significant, F(1, 50) = 4.18, p = .046, z = 1.99, observed power = 51%.
Once more, median observed power is just 57%, yet the success rate is 100%. Thus, the success rate is inflated by 43% and the R-Index is low, Ř = 14%, suggesting that an exact replication study will not produce significant results.
Studies 1 and 2 used highly selective samples (women in the top 10% in math performance). Study 3 aimed to replicate the results of Study 2 in a less selective sample. One might expect that stereotype-threat has a weaker effect on math performance in this sample because stereotype threat can undermine performance when ability is high, but anxiety is not a factor in performance when ability is low. Thus, Study 3 is expected to yield a weaker effect and a larger sample size would be needed to demonstrate the effect. However, sample size was approximately the same as in Study 2 (36 women, 31 men).
The ANOVA showed a main effect of sex on math performance, F(1, 63) = 6.44, p = .014, z = 2.47, observed power = 69%.
The ANOVA also showed a significant interaction between sex and stereotype-threat-assurance, F(1, 63) = 4.78, p = .033, z = 2.14, observed power = 57%.
Once more, the R-Index is low, Ř = 26 (MOP = 63%, Success Rate = 100%, Inflation Rate = 37%).
The three studies reported six statistical tests. The R-Index for the combined analysis is low Ř = 18 (MOP = 59%, Success Rate = 100%, Inflation Rate = 41%).
The probability of this event to occur by chance can be assessed with the Test of Insufficient Variance (TIVA). TIVA tests the hypothesis that the variance in p-values, converted into z-scores, is less than 1. A variance of one is expected in a set of exact replication studies with fixed true power. Less variance suggests that the z-scores are not a representative sample of independent test scores. The variance of the six z-scores is low, Var(z) = .04, p < .001, 1 / 1309.
Correction: I initially reported, “A chi-square test shows that the probability of this event is less than 1 out of 1,000,000,000,000,000, chi-square (df = 5) = 105.”
I made a mistake in the computation of the probability. When I developed TIVA, I confused the numerator and denominator in the test. I was thrilled that the test was so powerful and happy to report the result in bold, but it is incorrect. A small sample of six z-scores cannot produce such low p-values.
The replicability analysis of Spencer, Steele, and Quinn (1999) suggests that the original data provided inflated estimates of effect sizes and replicability. Thus, the R-Index predicts that exact replication studies would fail to replicate the effect.
A forthcoming article in the Journal of School Psychology reports the results of a meta-analysis of stereotype-threat studies in applied school settings (Flore & Wicherts, 2014). The meta-analysis was based on 47 comparisons of girls with stereotype threat versus girls without stereotype threat. The abstract concludes that stereotype threat in this population is a statistically reliable, but small effect (d = .22). However, the authors also noted signs of publication bias. As publication bias inflates effect sizes, the true effect size is likely to be even smaller than the uncorrected estimate of .22.
The article also reports that the after a correction for bias, using the trim-and-fill method, the estimated effect size is d = .07 and not significantly different from zero. Thus, the meta-analysis reveals that there is no replicable evidence for stereotype-threat effects on schoolgirls’ math performance. The meta-analysis also implies that any true effect of stereotype threat is likely to be small (d < .2). With a true effect size of d = .2, the original studies by Steel et al. (1999) and most replication studies had insufficient power to demonstrate stereotype threat effects, even if the effect exists. A priori power analysis with d = .2 would suggest that 788 participants are needed to have an 80% chance to obtain a significant result if the true effect is d = .2. Thus, future research on this topic is futile unless statistical power is increased by increasing sample sizes or by using more powerful designs that can demonstrate small effects in smaller samples.
One possibility is that the existing studies vary in quality and that good studies showed the effect reliably, whereas bad studies failed to show the effect. To test this hypothesis, it is possible to select studies from a meta-analysis with the goal to maximize the R-Index. The best chance to obtain a high R-Index is to focus on studies with large sample sizes because statistical power increases with sample size. However, the table below shows that there are only 8 studies with more than 100 participants and the success rate in these studies is 13% (1 out of 8), which is consistent with the median observed power in these studies 12%.
It is also possible to select studies that produced significant results (z > 1.96). Of course, this set of studies is biased, but the R-Index corrects for bias. If these studies were successful because they had sufficient power to demonstrate effects, the R-Index would be greater than 50%. However, the R-Index is only 49%.
In conclusion, a replicability analysis with the R-Index shows that stereotype-threat is an elusive phenomenon. Even large replication studies with hundreds of participants were unable to provide evidence for an effect that appeared to be a robust effect in the original article. The R-Index of the meta-analysis by Flore and Wicherts corroborates concerns that the importance of stereotype-threat as an explanation for gender differences in math performance has been exaggerated. Similarly, Ganley, Mingle, Ryan, Ryan, and Vasilyeva (2013) found no evidence for stereotype threat effects in studies with 931 students and suggested that “these results raise the possibility that stereotype threat may not be the cause of gender differences in mathematics performance prior to college.” (p 1995).
The main novel contribution of this post is to reveal that this disappointing outcome was predicted on the basis of the empirical results reported in the original article by Spencer et al. (1999). The article suggested that stereotype threat is a pervasive phenomenon that explains gender differences in math performance. However, The R-Index and the insufficient variance in statistical results suggest that the reported results were biased and, overestimated the effect size of stereotype threat. The R-Index corrects for this bias and correctly predicts that replication studies will often result in non-significant results. The meta-analysis confirms this prediction.
In sum, the main conclusions that one can draw from 15 years of stereotype-threat research is that (a) the real reasons for gender differences in math performance are still unknown, (b) resources have been wasted in the pursuit of a negligible factor that may contribute to gender differences in math performance under very specific circumstances, and (c) that the R-Index could have prevented the irrational exuberance about stereotype-threat as a simple solution to an important social issue.
In a personal communication Dr. Spencer suggested that studies not included in the meta-analysis might produce different results. I suggested that Dr. Spencer provides a list of studies that provide empirical support for the hypothesis. A year later, Dr. Spencer has not provided any new evidence that provides credible evidence for stereotype-effects. At present, the existing evidence suggests that published studies provide inflated estimates of the replicability and importance of the effect.
This blog also provides further evidence that male and female psychologists could benefit from a better education in statistics and research methods to avoid wasting resources in the pursuit of false-positive results.
Stanley and Doucouliagos (2013) demonstrated how meta-regression can be used to obtain unbiased estimates of effect sizes from a biased set of original studies. The regression approach relies on the fact that small samples often need luck or questionable practices to produce significant results, whereas large samples can show true effects without the help of luck and questionable practices. If questionable practices or publication bias are present, effect sizes in small samples are inflated and this bias is evident in a regression of effect sizes on sampling error. When bias is present, the intercept of the regression equation can provide a better estimate of the average effect size in a set of studies.
One limitation of this approach is that other factors can also produce a correlation between effect size and sampling error. Another problem is that the regression equation can only approximate the effect of bias on effect size estimates.
The R-Index can complement meta-regression in several ways. First, it can be used to examine whether a correlation between effect size and sampling error reflects bias. If small samples have higher effect sizes due to bias, they should also yield more significant results than the power of these studies justifies. If this is not the case, the correlation may simply show that smaller samples examined stronger effects. Second, the R-Index can be used as an alternative way to estimate unbiased effect sizes that does not rely on the relationship between sample size and effect size.
The usefulness of the R-Index is illustrated with Stanley and Doucouliagos (2013) meta-analysis of the effectiveness of nicotine replacement therapy (the patch). Table A1 lists sampling errors and t-values of 42 studies. Stanley and Doucouliagos (2013) found that the 42 studies suggested a reduction in smoking by 93%, but that effectiveness decreased to 22% in a regression that controlled for biased reporting of results. This suggests that published studies inflate the true effect by more than 300%.
I entered the t-values and standard errors into the R-Index spreadsheet. I used sampling error to estimate sample sizes and degrees of freedom (2 / sqrt [N]). I used one-tailed t-tests to allow for negative t-values because the sign of effects is known in a meta-analysis of studies that try to show treatment effects. Significance was tested using p = .025, which is equivalent to using .050 in the test of significance for two-tailed tests (z > 1.96).
The R-Index for all 42 studies was 27%. The low R-Index was mostly explained by the low power of studies with small samples. Median observed power was just 34%. The number of significant results was only slightly higher 40%. The inflation rate was only 7%.
As studies with low power add mostly noise, Stanley (2010) showed that it can be preferable to exclude them from estimates of actual effect sizes. The problem is that it is difficult to find a principled way to determine which studies should be included or excluded. One solution is to retain only studies with large samples. The problem with this approach is that this often limits a meta-analysis to a small set of studies.
One solution is to compute the R-Index for different sets of studies and to base conclusions on the largest unbiased set of studies. For the 42 studies of nicotine replacement therapy, the following effect size estimates were obtained (effect sizes are d-values, d = t * se).
The results show the highest R-Index for studies with more than 80 participants. For these studies, observed power is 83% and the percentage of significant results is also 83%, suggesting that this set of studies is an unbiased sample of studies. The weighted average effect size for this set of studies is d = .44. The results also show that the weighted average effect size does not change much as a function of the selection of studies. When all studies are included, there is evidence of bias (8% inflation) and the weighted average effect size is inflated, but the amount of inflation is small (d = .56 vs. d = .44, difference d = .12).
The small amount of bias appears to be inconsistent with Stanley and Doucouliagos (2013) estimate that an uncorrected meta-analysis overestimates the true effect size by over 300% (93% vs. 22% RR). I therefore also examined the log(RR) values in Table 1a.
The average is .68 (compared to the simple mean reported as .66); the median is .53 and the weighted average is .49. The regression-corrected estimate reported by Stanley and Doucouliagos (2013) is .31. The weighted mean for studies with more than 80 participants is .43. It is now clear why Stanley and Doucouliagos (2013) reported a large effect of the bias correction. First, they used the simple mean as a comparison standard (.68 vs. 31). The effect would be smaller if they had used the weighted mean as a comparison standard (.49 vs. .31). Another factor is that the regression procedure produces a lower estimate than the R-Index approach (.31 vs. 43). More research is needed to compare these results, but the R-Index has a simple logic. When there is no evidence of bias, the weighted average provides a reasonable estimate of the true effect size.
Stanley and Doucouliagos (2013) used regression of effect sizes on sampling error to reveal biases and to obtain an unbiased estimate of the typical effect size in a set of studies. This approach provides a useful tool in the fight against biased reporting of research results. One limitation of this approach is that other factors can produce a correlation between sampling error and effect size. The R-Index can be used to examine how much reporting biases contribute to this correlation. The R-Index can also be used to obtain an unbiased estimate of effect size by computing a weighted average for a select set of studies with a high R-Index.
A meta-analysis of 42 studies of nicotine replacement theory illustrates this approach. The R-Index for the full set of studies was low (24%). This reveals that many studies had low power to demonstrate an effect. These studies provide little information about effectiveness because non-significant results are just as likely to be type-II errors as demonstrations of low effectiveness.
The R-Index increased when studies with larger samples were selected. The maximum R-Index was obtained for studies with at least 80 participants. In this case, observed power was above 80% and there was no evidence of bias. The weighted average effect size for this set of studies was only slightly lower than the weighted average effect size for all studies (log(RR) = .43 vs. .49, RR = 54% vs. 63%, respectively). This finding suggests that smokers who use a nicotine patch are about 50% more likely to quit smoking than smokers without a nicotine patch.
The estimate of 50% risk reduction challenges Stanley and Doucouliagos’s (2013) preferred estimate that bias correction “reduces the efficacy of the patch to only 22%.” The R-Index suggests that this bias-corrected estimate is itself biased.
Another important conclusion is that studies with low power are wasteful and uninformative. They generate a lot of noise and are likely to be systematically biased and they contribute little to a meta-analysis that weights studies by sample size. The best estimate of effect size was based on only 6 out of 42 studies. Researchers should not conduct studies with low power and editors should not publish studies with low power.