Antivaccine nonsense Bad science Medicine

Antivax lies about vaccines and infant mortality, 12 years later

Since COVID-19, in the antivax world everything old is new again. That’s right, Gary S. Goldman and Neil Z. Miller are back to defend their 2011 infant mortality “study” and RFK Jr. is flogging it as slam-dunk “evidence” that vaccines kill babies.

I’ve lost track of the number of times that I’ve written this in the three years since a novel coronavirus causing a deadly respiratory disease that came to be named COVID-19 started spreading from Asia to Europe and the rest of the world. Unfortunately, however, yet another opportunity has arisen to say it again. In the world of antivaccine misinformation, disinformation, conspiracy theories, and pseudoscience, everything old is new again. Antivaccine misinformation that I first encountered a decade—or even two decades!—ago has been reborn and repurposed to attack COVID-19 vaccines, while the fear of COVID-19 vaccines has led previously COVID-19 vaccine-only antivaxxers to embrace all manner of old antivaccine misinformation about vaccines other than COVID-19 vaccines. It’s an amplification loop in which old techniques of demonizing vaccines applied to COVID-19 that seem new because no one other than antivaxxers and those of us who were paying attention to antivaxxers before the pandemic had encountered it metastasize back to affect all childhood vaccines again, thus fomenting a more general vaccine hesitancy for all vaccines, even among people who would have considered themselves “pro-vaccine” before the pandemic. In brief, “new school” COVID-19 antivaxxers are increasingly becoming indistinguishable from “old school” antivaxxers. That’s why it should be no surprise that antivaxxers are taking full advantage in order to use COVID-19 vaccine mandates to attack all vaccine mandates, including school mandates. Antivaxxers are also using distrust of COVID-19 vaccines to promote general distrust of all vaccines, especially childhood vaccines

So it was with a combination of alarm and amusement that I came across an article on Robert F. Kennedy, Jr.’s antivaccine website Children’s Health Defense entitled Higher Infant Mortality Rates Linked to Higher Number of Vaccine Doses, New Study Confirms. One of the advantages of having been studying antivaccine misinformation, pseudoscience, and conspiracy theories is that the headline itself made me think that I knew what this was and who was behind this “study.” It turns out that my suspicions were correct.

Everything old is new again: Prelude

First, here’s how RFK Jr.’s minion Michael Nevradakis spins it in his story:

A new peer-reviewed study found a positive statistical correlation between infant mortality rates (IMRs) and the number of vaccine doses received by babies — confirming findings made by the same researchers a decade ago. In “Reaffirming a Positive Correlation Between Number of Vaccine Doses and Infant Mortality Rates: A Response to Critics,” published Feb. 2 in Cureus, authors Gary S. Goldman, Ph.D., an independent computer scientist, and Neil Z. Miller, a medical researcher, examined this potential correlation. Their findings indicate a “positive correlation between the number of vaccine doses and IMRs is detectable in the most highly developed nations.” The authors replicated the results of a 2011 statistical analysis they conducted, and refuted the results of a recent paper that questioned those findings.

Bingo. My suspicion was correct. This was the study, published in a journal that many of you might recognize, Human & Experimental Toxicology, which has been a font of bad antivax studies dating back years and years, entitled Infant mortality rates regressed against number of vaccine doses routinely given: Is there a biochemical or synergistic toxicity? And guess what? I deconstructed how awful that study was in 2011, shortly after it had been published. Regular readers know that I can rarely pass up a chance to revisit old disinformation resurrected and tarted up by antivaxxers to frighten people about vaccines yet again, and so as soon as I knew that this was Goldman and Miller resurrected I knew I had to write about this again because I had seen it bubbling up on social media, thanks to RFK Jr., for example:

I included that response, because it’s been a longstanding antivax trope that vaccines cause sudden infant death syndrome (SIDS). They don’t.

Another example:

Of course, one question that immediately occurred to me was: Why now? Why is this old antivax message that vaccines kill babies popping up again now? I think this is the answer:

I’ve seen rants about this all over Twitter last week. Although Goldman and Miller likely couldn’t have known that the CDC might add COVID-19 vaccines to the recommended routine childhood vaccination schedule now, they could have been looking for another opportunity to promote their message. On the other hand, I looked at the journal website for Cureus, which is the journal where Goldman and Miller published their “new” study on vaccines and childhood mortality rates, and saw that the journal touts its average time to first decision as 1.7 days and its time to first publication as 37 days. These are pretty fast. This journal has also published some very dodgy work on ivermectin touting it as highly effective prophylaxis against COVID-19. (It’s not, as I discussed.)

So let’s compare Goldman and Miller in 2011 to Goldman and Miller in 2023. Here’s RFK Jr.’s minion Nevradakis again:

In 2011, Miller and Goldman published a peer-reviewed study in Human and Experimental Toxicology, which first identified a positive statistical correlation between IMRs and number of vaccine doses. The researchers wrote:

“The infant mortality rate (IMR) is one of the most important indicators of the socio-economic well-being and public health conditions of a country. The U.S. childhood immunization schedule specifies 26 vaccine doses for infants aged less than 1 year — the most in the world — yet 33 nations have lower IMRs. “Using linear regression, the immunization schedules of these 34 nations were examined and a correlation coefficient of r = 0.70 (p < 0.0001) was found between IMRs and the number of vaccine doses routinely given to infants.”

In the above figures, “r” refers to the correlation coefficient, a number that ranges from -1 to 1. Any figure above zero is understood as a positive correlation, with figures between 0.6 and 0.79 considered a “strong” positive correlation, and 0.8 and above a “very strong” positive correlation. The “p-value” indicates the extent to which the predictor’s value, in a linear regression analysis, is related to changes in the response variable. A p-value of 0.05 or below is considered statistically significant, and indicative that the predictor and the response variable are related to each other and move in the same direction. In the same 2011 study, which used 2009 data, the researchers found that developed nations administering the most vaccine doses to infants (21 to 26 doses) tended to have the worst IMRs.

I can hear epidemiologists and statisticians facepalming in unison at the very concept of doing a linear regression between vaccine doses in the childhood vaccine schedule of different nations and infant mortality rates (IMRs) in those countries. That’s before you even consider how completely arbitrary and riddled with errors the method used to estimate the number of “vaccines” in each nation’s recommended schedule was, as I’ll quote and recap from 12 years ago:

Arbitrary: they count number of vaccines in US bins (DTaP is one, hib is separate) and non-specific designations (some “polio” is still given as OPV in Singapore), rather than antigens. If they did that, Japan, still giving the live bacterial vaccine BCG, would immediately go to the top of the list. That wouldn’t fit the agenda, of course. But if you go by “shot” rather than by antigen, why are DTaP, IPV, hepB and hib counted as 4 shots for example in Austria, when they are given as Infanrix hexa, in one syringe? Mistakes: The German childhood vaccination schedule recommends DTaP, hib, IPV AND hepB, as well as PCV at 2, 3 and 4 months, putting them squarely into the 21 – 23 bin. The fourth round of shots is recommended at 11 to 14 months, and MenC, MMR and Varicella are recommended with a lower age limit of 11 months, too, which means that a number of German kids will fall into the highest bin, at least as long as you count the Miller/Goldman way. Then, they neatly put those arbitrarily counted doses into bins. Binning (i.e. grouping numbers before correlating them to something) always makes me suspicious. I don’t have the time to check each country’s vaccination schedule – I assume there will be mistakes in many claims, but I am guessing that if we plotted the infant mortality against the actual number of recommended vaccines, the correlation would be less good than engineered in this paper, i.e. the dose count above is probably not all that “arbitrary”.

In brief, Goldman and Miller arbitrarily—but really, not quite so arbitrarily—defined multivalent vaccines into multiple vaccine doses as it suited them to promote their message, used a method that epidemiologists would laugh at given that it is the ecological fallacy on steroids, and then didn’t even try to control for confounding factors, of which there were many. Also, did I mention that they cherry picked only 34 countries and then removed four nations, Andorra, Liechenstein, Monaco, and San Marino, the justification being that because they are all so small, each nation only recorded less than five infant deaths? (One wonders whether including them would have made the “correlation” less striking.) Or that, more dubiously, for some reason the authors, not content with an weak and not particularly convincing linear relationship in the raw data, decided to do a little creative data manipulation and divide the nations into five groups based on number of vaccine doses, take the means of each of these groups, and then regraph the data? Not surprisingly, the data look a lot cleaner, which was no doubt why this was done, as it was a completely extraneous analysis.

Then, as I noted then, it’s very dicey to compare IMRs across nations because different countries define an infant mortality, differently, and I cited an article describing why that is no longer available even on the Wayback Machine at and will quote this passage again:

First, it’s shaky ground to compare U.S. infant mortality with reports from other countries. The United States counts all births as live if they show any sign of life, regardless of prematurity or size. This includes what many other countries report as stillbirths. In Austria and Germany, fetal weight must be at least 500 grams (1 pound) to count as a live birth; in other parts of Europe, such as Switzerland, the fetus must be at least 30 centimeters (12 inches) long. In Belgium and France, births at less than 26 weeks of pregnancy are registered as lifeless. And some countries don’t reliably register babies who die within the first 24 hours of birth. Thus, the United States is sure to report higher infant mortality rates. For this very reason, the Organization for Economic Cooperation and Development, which collects the European numbers, warns of head-to-head comparisons by country. Infant mortality in developed countries is not about healthy babies dying of treatable conditions as in the past. Most of the infants we lose today are born critically ill, and 40 percent die within the first day of life. The major causes are low birth weight and prematurity, and congenital malformations. As Nicholas Eberstadt, a scholar at the American Enterprise Institute, points out, Norway, which has one of the lowest infant mortality rates, shows no better infant survival than the United States when you factor in weight at birth.

I will also cite this one that explains why US IMR appears so high compared to other countries. (Hint: It has nothing to do with the number of vaccines in the childhood vaccination schedule.)

Fast forward 12 years.

Everything old is new again: 2023 Goldman and Miller edition

As the adage goes, when your only tool is a hammer, everything starts looking like nails, and I can’t think of a better example to which to apply this adage than Goldman and Miller. Basically, in this new paper, they double down on their previous linear regressions and then try to do additional questionable analyses to support their previous analysis. Also, I get the feeling from their introduction that they’re really, really cheesed at a preprint published in 2021 and updated through 2022 that debunked their 2011 “study.” To be honest, I was puzzled when I learned of this article, mainly because I wondered why anyone would bother in 2021 (when the first version was posted to a preprint server) to rehash a 2011 antivax paper. Helpfully, the authors of the preprint, Nysetvold et al, explain:

Although the 2011 study was published in a peer-reviewed journal (Human and Experimental Toxicology), a brief reading of the Miller and Goldman manuscript led us to question the methods, results and conclusions. We observed significant deficiency in the statistical methods. Thus, it is troublesome that this manuscript is in the top 5% of all research outputs since its publication, being shared extensively on social media with tens of thousands of likes and re-shares (see

My fault is that I hadn’t realized that this paper had been so widely shared and cited in the last 12 years; I should have. I basically debunked the paper as ridiculous and then moved on to the many, many other bad antivax studies being published. First, though, let’s look at the preprint’s conclusions. Being actual scientists, the team led by Elizabeth Bailey at Brigham Young University found, using the same data used by Goldman and Miller, plus data for all 185 nations, the following:

We show that the sub-sample of 30 countries used in the original paper was an unlikely random sample from the entire dataset, as the correlation coefficient of 0.49 reported in that study would only arise about 1 in 100,000 times from random sampling. If we investigate only countries with high or very high development, human development index explains the variability in IMR, and vaccine dose number does not. Next, we show IMR as a function of countries’ actual vaccination rates, rather than vaccination schedule, and show a strong negative correlation between vaccination rates and IMR. Finally, we analyze United States IMR data as a function of Hepatitis B vaccination rate to show an example of increased vaccination rates corresponding with reduced infant death over time. From our analyses, it is clear that vaccination does not predict higher IMR as previously reported.

Quelle surprise. Goldman and Miller’s findings were easily explained by actual scientists as arising primarily from a nonrandom selection of nations, not looking at actual vaccination rates rather than the suggested vaccine schedule, and the human development index of the nations examined as a major confounder. That reminds me that I should have really pointed out before now that it is a fallacy to compare just the vaccine schedule to IMRs because just because a vaccine is on the schedule doesn’t mean that children are getting those vaccines, given the variation in resources and health care availability between nations—and even within nations. Another thing that I should have questioned is the very assumption that any correlation between the number of vaccine doses and IMRs must be linear. There is no a priori reason to assume that, which makes Miller and Goldman’s insistence on doing linear regression after linear regression dubious to begin with. Indeed, Bailey’s group notes this:

We checked the assumptions of linear regression for each regression analysis (results in each respective figure or table folder, see also file in github repository for details). For all analyses except rotavirus in Figure 3, the assumption of homoscedasticity was violated. Thus, we used the vcovHC function from the sandwich package in R to generate heteroscedasticity robust standard errors for our hypothesis testing (36). Otherwise, all assumptions of linear regression were met.

Homoscedasticity is defined thusly:

The assumption of homoscedasticity (meaning “same variance”) is central to linear regression models. Homoscedasticity describes a situation in which the error term (that is, the “noise” or random disturbance in the relationship between the independent variables and the dependent variable) is the same across all values of the independent variables. Heteroscedasticity (the violation of homoscedasticity) is present when the size of the error term differs across values of an independent variable. The impact of violating the assumption of homoscedasticity is a matter of degree, increasing as heteroscedasticity increases.

None of this should be surprising given how different the data sources being compared are. So I am going to fast-forward to Goldman and Miller’s discussion, where they state:

The Bailey reanalysis contains heteroscedastic data. Due to the dissimilar values of IMR over the range of the number of vaccine doses, Bailey’s data do not have a constant variance (homoscedasticity). This leads to less precise coefficient estimates, and statistical tests of significance are invalidated.

What Goldman and Miller fail to mention is that Bailey’s team did an appropriate mathematical calculation in order to be able to examine heteroscedastic data. They just make it sound as though Bailey just ignorantly plugged heteroscedastic data into their linear regression without further analysis, not unlike the way they did in their first paper. (Projection is epic among antivaxxers.)

In fact, Goldman and Miller give the game away in their introduction when they worry that Bailey and her team “appear to be targeting our study for a potential retraction.” My first reaction to this lament is, “Gee, you say that as though it were a bad thing.” After all, Goldman and Miller’s study should never have been published in a peer-reviewed journal; indeed, it should have been retracted a decade ago. However, the journal in which it was published has long been a haven for poorly designed and executed antivaccine studies; so getting its editors to retract anything is not particularly likely.

A particularly telling part of Goldman and Miller’s paper is its complaint that Bailey’s group looked at 185 countries and therefore included outliers. What they ignore is that Bailey’s group looked at all of those groups not just to test to see if the Miller/Goldman “analysis” would hold up for a much larger sample of countries, but to test how likely their result would be using a truly random subset of the 185 nations:

In order to better visualize how extreme Miller and Goldman’s result was even within their own dataset, we randomly sampled 30 countries from the full dataset of 185 countries and computed the linear regression. This sampling was done 50,000 times, and the distribution of regression results was plotted (Figure 2). We then determined the degree to which Miller and Goldman’s result (R2 = 0.493) may be considered an outlier. Within this distribution of random samples, the mean R2 was 0.049 with a standard deviation of 0.053. We calculated the z-score of 0.493 against our distribution to be 8.3, meaning there is approximately a 1 in a 100,000 chance that this result was achieved with a random sample of the dataset. To verify this, we performed 1 million random samplings, and the most extreme R2 observed was 0.577, with only 10 samples’ R2 exceeding 0.493. Therefore, we conclude that the sample of 30 countries from the Miller and Goldman analysis is not representative of the full dataset.

The bottom line? Goldman and Miller did indeed cherry pick the countries they examined.

So what about the new “analysis” by Goldman and Miller? Well, including 185 countries doesn’t yield much of a better result:

Infant mortality graph
This is what we in the biz call a “star chart.” Notice how the points removed from the graph on the left are all mainly the ones clustered around very low infant mortality rates.

Basically, this is what we in the biz call a “star chart.” For one thing, an r=0.16 is basically no significant correlation. In fact, r2 (which is what we are really interested in) would be 0.0256. So basically, even taking Goldman and Miller’s result at face value, vaccine dose number would only explain 2.6% of the variance in the model using all 185 nations, and they couldn’t let that result stand, as even they appear to sense that what they are showing is basically noise, particularly when you consider the lack of control for confounders. Sure, there’s a p-value that is (barely) under 0.05, but this is still basically an r-value that is indistinguishable from zero or so small as to be clinically meaningless, either at a population level or at an individual level. Indeed, if I ever found a correlation coefficient that low in a proposed linear regression model, I’d either conclude that there is no linear relationship between the variables or start looking for more complex models. In this case, I’d conclude the former.

Miller and Goldman appear to recognize this, at least implicitly, as they start slicing and dicing the dataset to try to find more—shall we say?—convincing r-values. Interestingly, to this scientist at least, they do the exact opposite. Their sensitivity analysis, in which they add more countries to their original 30 nations from their 2011 paper shows something very telling:

Infant mortality graph #2
This is a case where a graph doesn’t show what the authors think it does. Also, even if r=0.70, that would mean r2=0.49, which is not a high number for the variance attributable to the number of vaccines in the vaccine schedule in a model like this.

In brief, Goldman and Miller added about infant mortality:

Next, using IMRs and vaccine doses derived from the same sources, linear regression analyses continued to be performed successively on each of the top 31, 32, 33,…. nations until the reported r-value was no longer statistically significant (at n = 47). The correlation coefficient (r) decreased from r = 0.70 (p < .0001) for n = 30 to r = 0.29 (p = .0504) for n = 47, as shown in Figure 3 and Table 3.


While our original study analyzed the top 30 nations with the US as the cutoff, an additional 16 nations (from Croatia #31 to Russia #46) could have been included in the linear regression of IMRs vs. the number of vaccine doses, and the findings would still have yielded a statistically significant positive correlation coefficient.

They present this as a test for their assumptions, but in reality what this graph really suggests is just how biased their original sample of 30 countries was, given that as they added nations one-by-one, their correlation coefficient got successively worse. Again, as we like to point out, “statistically significant” does not necessarily mean “significant,” particularly for datasets like this, no matter how much this not-so-dynamic duo want to blame vaccines for infant mortality.

The last bit of Goldman and Miller’s “rebuttal” is a lot of projection:

The Bailey team indiscriminately combined highly developed and Third World nations without regard to confounding arising from heterogeneous data. Clusters of heterogeneous nations (due to variable socioeconomic factors) report widely differing IMRs yet prescribe the same number of vaccine doses for their infants. For example, Angola (with an IMR of 180.21) and Belgium (with an IMR of 4.44) both require 22 vaccine doses.

If Goldman and Miller weren’t antivaccine hacks, this observation should tell them something very important, namely that differences in IMR depend on many things and that their results don’t show an effect that can be correlated with the number of vaccine doses on their recommended vaccine schedule.

Basically, Miller and Goldman double down on their previous analysis and then handwave all sorts of complaints about the Bailey analysis. For example, they try to explain away Bailey’s finding that the results they got from their 30 nation subset of the 185 nations in the dataset were incredibly unlikely to result from a random sampling by stating that the “analysis is flawed because it mixed nations displaying heterogeneity of socioeconomic factors without regard to covariates.” This is the pot calling the kettle black given that the original Goldman and Miller analysis made nothing resembling a statistically legitimate attempt to control for confounders and covariates and the entire point of the Bailey analysis was to show how doing a truly random sample would be incredibly unlikely to generate the results that Goldman and Miller got.

Many of the rest of Goldman and Miller’s complaints are nitpicking in that what they cited as “deficiencies” would be incredibly unlikely to have converted an analysis implicating higher numbers of vaccines in a nation’s childhood vaccination schedule as not correlating with increasing IMRs to one that does. Indeed, as Bailey’s group shows, the opposite correlation, in which vaccines are protective against IMR (which is what has been known for a very long time), is far more likely to be the correct one.

Goldman and Miller also criticize Bailey for counting vaccines correctly (yes, I’m being sarcastic):

The number of vaccine doses for some nations is substantially miscounted. For example, seven infant vaccine doses are reported for Australia when the true value is 22 (as indicated by WHO/UNICEF) or 24 (as indicated by the immunization schedule published by the Australian government). Incorrectly calculated vaccine doses, whether due to the exclusion of subnationally distributed vaccines or a manual miscount, are problematic.

I quite literally laughed out loud when I read that passage, as I did with all the other anomaly hunting that Goldman and Miller did in nitpicking oddities in Bailey’s results. If I wanted to get Biblical, I could cite Matthew 7:3-5: “And why beholdest thou the mote that is in thy brother’s eye, but considerest not the beam that is in thine own eye?” (It is very rare that I quote the King James version of The Bible—or even The Bible in general—for anything, but it seems appropriate here.)

Finally, near the very end, Goldman and Miller start handwaving about “biological mechanisms” (that aren’t based in any actual biology), in order to make their antivaccine propaganda sound more compelling. For example, they pull out the hoary old antivax chestnut falsely blaming vaccines for SIDS, claiming that there “is credible evidence that a subset of infants may be at increased risk of sudden infant death shortly after being vaccinated.” (Again, there ain’t; if anything, the opposite is true and vaccines are likely protective against SIDS. At the very worst, they have no detectable effect on SIDS risk.) This not-so-dynamic duo even cites a Vaccine Court ruling:

On July 10, 2017, the US Court of Federal Claims [28] issued a decision with regard to a claim filed with the Vaccine Injury Compensation Program (VICP). A male infant, J.B., received seven vaccines at his four-month well-baby visit. On the following day, he died. The medical examiner stated that the cause of death was SIDS. His parents filed a petition under the VICP. Petitioners allege that as a result of receiving vaccines for diphtheria, tetanus, acellular pertussis, polio, Hib, pneumococcal, and rotavirus, J.B. passed away from SIDS. After listening to expert testimony by Dr. Douglas C. Miller, a neuropathologist, Special Master Thomas L. Gowen concluded that the petitioners “have demonstrated by a preponderance of the evidence that the vaccines can and likely did play a critical role in this child’s death.”

Had I been a peer reviewer for this paper, I would have pointed out that the Vaccine Court sometimes makes mistakes (as it did in this case) and that the ruling was ultimately overturned, something Goldman and Miller somehow fail to mention. I wonder why.

Actually, I don’t. Miller and Goldman are antivaccine hacks. They’ve been publishing “studies” that erroneously come to the conclusion that vaccines are harmful for a very long time. Indeed, Gary Goldman states in his Cureus bio—wait, they have bios on Cureus?—that he is not a medical professional, and it shows. The same is true for Neil Miller. Apparently someone who contributed to this study, Walter Schumm, is listed as a statistician. Although it’s claimed that Schumm did an odds-ratio analysis, he is not credited as a co-author on this paper and is not even listed in the acknowledgments. More disturbingly, although the paper claims that Schumm did an odds-ratio analysis “whereby nations in our study were divided at the median IMR and total vaccine doses, controlling for several confounding variables (including low birth weight, child poverty, and breast feeding)” and found that none “of these factors lowered the original correlation below 0.62.” Of course, given that the original correlation of 0.70 was based on a clearly nonrandom and biased selection of nations, this additional analysis is basically the proverbial putting lipstick on a pig. It doesn’t change the essential lack of scientific validity of the original analysis.

Never having heard of Schumm before, I Googled and found that he is an emeritus professor in the Department of Applied Human Sciences at Kansas State University. I also note that he does not appear to have an advanced degree in statistics (or any degree in statistics), although he does still apparently teach basic statistics. To properly do the analyses claimed would require a bit more—shall we say?—advanced statistical skills. In any event, the best that I can say about Schumm after doing some searches about him is that he appears to possess expertise in social science research methodology, but there’s no good evidence that he has expertise to do an analysis like Goldman and Miller’s. A PubMed search on his name also reveals studies like this one concluding that “psupport of transgender children may temporarily reduce levels of poor mental health for some transgender children, but it does not appear to eliminate those problems for all transgender children” (as if anyone claims that gender-affirming care is effective at this 100% of the time). Then there was this one, in which Schumm claims to have found a “pro-homosexual bias” in the social science literature, and then in this letter he co-authored with Andre Von Mol in which he questioned research finding adverse health outcomes associated with sexuality orientation change efforts (better known as “conversion therapy”).

The bottom line is that, once again, antivax hacks gonna hack, and Goldman and Miller are antivax hacks. Their 2011 study was pure hack, and their defense of that study in 2023 is more of the same. They’re not interested in science. Rather, above all they are interested in promoting a message claiming that vaccines kill children.

By Orac

Orac is the nom de blog of a humble surgeon/scientist who has an ego just big enough to delude himself that someone, somewhere might actually give a rodent's posterior about his copious verbal meanderings, but just barely small enough to admit to himself that few probably will. That surgeon is otherwise known as David Gorski.

That this particular surgeon has chosen his nom de blog based on a rather cranky and arrogant computer shaped like a clear box of blinking lights that he originally encountered when he became a fan of a 35 year old British SF television show whose special effects were renowned for their BBC/Doctor Who-style low budget look, but whose stories nonetheless resulted in some of the best, most innovative science fiction ever televised, should tell you nearly all that you need to know about Orac. (That, and the length of the preceding sentence.)

DISCLAIMER:: The various written meanderings here are the opinions of Orac and Orac alone, written on his own time. They should never be construed as representing the opinions of any other person or entity, especially Orac's cancer center, department of surgery, medical school, or university. Also note that Orac is nonpartisan; he is more than willing to criticize the statements of anyone, regardless of of political leanings, if that anyone advocates pseudoscience or quackery. Finally, medical commentary is not to be construed in any way as medical advice.

To contact Orac: [email protected]

66 replies on “Antivax lies about vaccines and infant mortality, 12 years later”

The fact that they even bothered with a linear regression on that set of 185 countries given that scatterplot speaks volumes about them. Also:

Another thing that I should have questioned is the very assumption that any correlation between the number of vaccine doses and IMRs must be linear. There is no a priori reason to assume that, which makes Miller and Goldman’s insistence on doing linear regression after linear regression dubious to begin with. Indeed, Bailey’s group notes this:

Correlation, as they’re using it, only applies to linear relationships: not linear? Programs will calculate correlations but there is no meaningful interpretation or valid meaning.

divide the nations into five groups based on number of vaccine doses, take the means of each of these groups, and then regraph the data

If they grouped based on the number of doses, with different groups having different ranges, averaging this way would [IMO] violate the requirement of constant variance of the response.

About this “objection”:

Bailey’s group looked at 185 countries and therefore included outliers. </blockquote.

Unless there is a valid reason [measurement error, they correspond to values recorded from the wrong target population, non-fixable data entry error] removing "outliers" simply becuase you view them as outliers is not good statistical practice. I'm not surprised folks in the anti-vaxx community don't know this.

The plot of correlation coefficients against number of nations is an epic self-own: not only [as you point out] do the correlations decrease against sample size, but they decrease by so much that they eventually become non-significant. Correlation tests are highly sensitive to sample size, so larger samples should

increase the detection of a significant correlation.

Poop: If I post this fixed version can the first one be canned? Very sorry.

Another thing that I should have questioned is the very assumption that any correlation between the number of vaccine doses and IMRs must be linear. There is no a priori reason to assume that, which makes Miller and Goldman’s insistence on doing linear regression after linear regression dubious to begin with. Indeed, Bailey’s group notes this:

Correlation, as they’re using it, only applies to linear relationships: not linear? Programs will calculate correlations but there is no meaningful interpretation.

divide the nations into five groups based on number of vaccine doses, take the means of each of these groups, and then regraph the data

If they grouped based on the number of doses, with different groups having different ranges, averaging this way would [IMO] violate the requirement of constant variance of the response.

About this “objection”:

Bailey’s group looked at 185 countries and therefore included outliers.

Unless there is a valid reason [measurement error, they correspond to values recorded from the wrong target population, non-fixable data entry error] removing “outliers” simply because you view them as outliers is not good statistical practice. I’m not surprised folks in the anti-vaxx community don’t know this.

The plot of correlation coefficients against number of nations is an epic self-own: not only [as you point out] do the correlations decrease against sample size, but they decrease by so much that they eventually become non-significant. Correlation tests are highly sensitive to sample size, so larger samples should increase the detection of a significant correlation.

“linear regression” does seem to be a preferred tool by the anti-vax ‘researchers’. Maybe they believe that the use of ‘mathematics’ will make their claims seem more scientific. Or maybe that’s the extent of their understanding…

It is easy to do and “understand” (often graphical.) it’s often the first method we teach, too.

It is often the one of the first items taught, but sadly it’s usually just the mechanics without any discussion of when it is appropriate, when it isn’t, and the things you need to be concerned about if you’re considering it.

The huge advance in available computing and software over the past couple decades has also been a double-edged sword: it’s helped us implement procedures that would have been hell back when I was in grad school, but it’s allowed people [like the anti vaccination cranks] to analyses without knowing what the hell they’re doing.

What difference will it make after the red states secede?**
-btw- Will parts of eastern Washington also secede to join Idaho?

I avoid red states ( visiting/ business, if I can) like the plague because……
( Complete my thought)

** as Taylor Greene hints.

It’s not passed in law yet, one could hope some sane minds may prevail.
OTOH, I will be my obnioxious self and state I’m very glad I’m not American.
Democracy is a wonderful thing, but if you keep electing people like that… It sorts of defeat the purpose.

More ignorant, political baloney that will only hurt their constituents. I see this like a bunch of people being angry about air nailers framing a building more quickly than a hammer. We get to the same outcome, just faster.

Where are the long term studies?! They demand. What if those air nails cause the 2x4s to ROT in 15 years?!! Rapid building failure has been seen with these nailers! We’re just asking questions about this scary new technology.

The reality is that most people understand molecular and genetic biology about as well as they understand structural engineering which is why this analogy is useful, in my mind.

If the next one that comes along, and there will be a next one, has a high case fatality rate or r0 or, worse-both, the mRNA vaccine technology is our only chance to field a vaccine in time to stop massive casualties.

I guess I can finally make big money off a vaccine at last! I’m not too far from their border. I’ll go there and set up a tent just across state lines.

@ Dr Yeti:

” What if those air-nails cause the 2x4s to ROT in 15 years?!!…”
That is hilarious.
True about scoffers’ grasp of bio being as bad as their grasp of structure engineering.

I thought you lived near Idaho. So you’ll observe the fiasco directly ( although you’ve probably already seen enough catastrophes)

re ” everything old is new again”

Reading/ hearing recent anti-vax material I recall that there were basically 3 different ways pseudoscientists portrayed how vaccines did their ‘devastating’ dirty work:
— mercury ( or other toxins)
— “too many, too soon”
— GI damage leading to brain damage
Even confabulation about mRNA vaccines involves some variant of the above.
While science advances, pseudoscience reiterates the same old ideas.

-btw- Orac writing about SIDS might provoke a particular anti-vaxxer’s habitual response

While science advances, pseudoscience reiterates the same old ideas.

That’s demonstrated by the fact that the original study Orac discusses uses an old trick by climate deniers: taking a set of data and carefully taking a small subset to analyze in order to “prove their point”.

Here Goldman and Miller used a sample that was unlikely random sample for some of their work. Climate deniers have routinely carefully selected small segments of large time series on climate and used those segments to support their claim that the climate isn’t changing.

Pete Seeger said that when Woody Guthrie was asked why so many folk songs sounded incredibly Guthrie responded

Plagiarism is common to all cultures

That seems to hold for the conspiracy minded folks as well as musicians.

Latest example* of antivax lies from Natural News: an article claiming that pancreatic cancer rates are “skyrocketing” because Covid vaccines and boosters are sending “toxic spike protein prions” into various organs and causing “turbo cancer”. The source for this claim is an unnamed paper published in Gastroenterology.

A likely reason they didn’t name or link to the paper is that it’s titled “Increasing Pancreatic Cancer Incidence in Young Women in the US: A Population-Based Time-Trend Analysis, 2001-2018”.

(there’s a hint in that title that Covid vaccines may not actually be responsible for what one of the paper’s authors is terming a small but concerning increase in pancreatic cancer, notably in women under 55)

The pandemic can rightly be linked to a markedly increased incidence of turbo lying.

*”betting there are newer examples of lies on the site by now.

there’s a hint in that title that Covid vaccines may not actually be responsible

Vaccines are so powerful, they bend the quantum space-time continununum and go harm people back in the past.

Vaccines are so powerful, they bend the quantum space-time continununum and go harm people back in the past.

Big Pharma has so much money it can rent Obama’ time machine.

One thing I learned years ago, its ‘the vaccine wot done it’, its always the vaccine.

Mike never stops lying/ misleading. Yesterday, he was admiring a religious revival, quoting scripture and predicting the messiah’s return. Also, banks will be shutting down on Friday, taking your money unless you accept digital currency. Like the other idiot I survey, he supports Putin over Zelenskyy. These guys sound like Tucker with green smoothie recipes.
Anti-trans, anti-woke, anti-vax, anti-racial history, anti- modernity, anti-education .. unless it’s their brand of education, i.e. abysmal

“Also, banks will be shutting down on Friday, taking your money unless you accept digital currency.”

Well that’s not the sign of an internet con man is it? LOL.

Mikey, whose conspiracy theories include currency collapse/ digital currency, food riots, contamination of most food supplies, destruction of the power grid/ cell network and gang war chaos/ nuclear war
sells – or accepts advertisement from companies that sell-
silver/ gold coins, storable meals/ supplements, seeds/ growing systems, satellite phone/ text , gun supplies, prepper supplies, iodine/ products for decontamination.

Ahhh… Sounds like the “Y2K DOOM” paranoid hysteria all over again…
Survivalist prep BS.
It’s teh bestest BS.
How many buckets of 20 year old beans and rice have become landfill for the gullible followers of these scammers?

As someone Ahem is wont to say – “What’s old is new again.”

Have fun.

@Reality Y2K was a real bug, and there were lots of bug fixing. Not the end of the world,of course.

New years eve 1999 was a nice little earner for me, as an IT support engineer I was drafted in to work incoming support calls at 10× base rate. Nothing happened (except ker ching).

Back when the Y2K worry was starting to be a big deal, my father (who was still working at a bank at the time) asked me what I thought the best investment would be in the run-up to Y2K. My response was ‘outsourced payroll support’, because frankly the systems most likely to have major issues would be the old back-office mainframes running the company payroll with software that had been written thirty years before and not touched since then, and those having problems would have massive economic ripple effects. Most of the rest of it was either going to be an annoyance at most (browsers showing ‘19100’ as dates instead of ‘2000’) or a non-issue because in general things like clocks don’t care about year or even date, only time.

Y2K was potentially a major issue. Once the panic started and pushed C-suite types to actually pay attention to the complaints their IT workers had been making for years already, and grant them the time and money to fix it, most of the potentially significant problems got fixed pretty quickly.

Of course, once the crisis was averted, people who had been panicking were disappointed that so little went wrong, and blamed the people who started the panic because ‘nothing happened’. Never mind that the people who actually fixed the problems would probably not have been given the chance to if the panic hadn’t started.

A hint?
I think the dates make it pretty obvious. Or can the vaccine travel back in time?

I am disappointed you left out the the information about Neil Z. Miller communicating with aliens. It is an essential piece of the puzzle when reviewing anything written by Miller.

Also missing, although this might be deemed a bit too obvious, is that the whole methodology is irredeemably flawed from the beginning. The majority of infant mortality occurs in the first month from birth (and a large component of that in the first week) except in the poorest of countries. Children are simply not receiving many vaccines in this period. It doesn’t matter if the schedule has 10 or 26 vaccines, most of these dead children will have received a maximum of 1.

Orac writes,

Notice how the points removed from the graph on the left are all mainly the ones clustered around very low infant mortality rates.

MJD says,

Furthermore, if n=30 in the Miller-Goldman study why are there only 25 points on the graph? Now will you release MJD from auto-moderation?

Furthermore, if n=30 in the Miller-Goldman study why are there only 25 points on the graph?

Hint. There isn’t.

What is needed to debunk this paper is not a point-by-point analysis of every flaw in it, but a robust study examining the relationship between the number of vaccines received with infant fatalities. These authors are trying to understand if it’s a reasonable attempt to test the hypothesis ‘too many, too soon’ without access to anything but country level data. The results are certainly not conclusive due to the inherent limitations of their dataset, but the results should be treated as a legitimate signal for further study.

The is information parents want to know with regard to assessing risk for their child. But their having access to that information does not increase industry profits. The data to run a stronger, more detailed, better planned and prepared study comparing those things most certainly exists. That’s what I need to see to convince me the proposed relationship is nothing. Show me better data that is able to strongly conclude that the correlation was spurious.

What is the proposed, plausible mechanism by which “Too many, too soon” causes pathology?
Why are you not worried about kids getting too many routine childhood infections too soon? You do realize they get an average of 6-9 URIs a year, right?
Why do vaccines, which do not involve actively-replicating, cell-damaging infections worry you and actively-replicating, cell-damaging infections don’t worry you?
Are you aware of the potential long-term damage that infections can do that vaccines absolutely cannot do?
Mounting evidence continues to implicate viral infections with all kinds of pathologies; the most concerning of which being various cancers. Would you like to run the risk that your kids get these infections and have a much higher risk of cancer or other disease later when we really start linking these things because you didn’t trust some imaginary boogeyman? That’ll be an interesting dinner conversation…

The results are certainly not conclusive due to the inherent limitations of their dataset, but the results should be treated as a legitimate signal for further study.

No they should not.

See my note above.

The whole exercise is flawed from the beginning as most infant mortality in developed countries occurs in the first month from birth when children have at most been given 1 vaccine – Hep B.

Even then, to get any sort of signal, Goldman and Miller had to cherry pick countries. As soon as other countries are added, the signal disappears. There was no signal to begin with. It was all anti-vaccine smoke, mirrors and lies.


In the US 2/3rds (66.3%) of infant deaths occur within 27 days of birth when, at most, 1 vaccine (Hep B) would have been administered.
Therefore it doesn’t matter if the 12 month infant vaccination schedule calls for 10 or 10 million vaccines since most deaths are occurring without vaccination.
There is no association between the two. Any correlation between the two is false and irrelevant.
This is just like there is no association between the autism rate and sales of organic food even though an irrelevant correlation between them can be shown.
There is no need for any ‘deep dive’ into the statistics of the data when it can be shown there is no association nor correlation between the data to start with.

Here’s the IMR vs Age at Infant Death data.
Source: CDC Wonder – Linked Birth / Infant Death Records, 2007-2020 Results
Age at Infant Death: Number of deaths (%) (Cumul. %)
Under 1 hour: 48,556 (14.6%) (14.6%)
1-23 hours: 86,071 (25.8%) (40.4%)
1-6 days: 43,322 (13.0%) (53.4%)b
7-27 days: 43,265 (13.0%) (66.3%)
28-364 days: 112,263 (33.7%) (100.0%)
Total: 333,477

See these posts:
Screenshot of the above CDC Wonder data:
Another quick takedown of this foolishness is the chart of US IMR verus Year:
In 1960 when the infant vaccination schedule was very low the IMR was ~27 and in 2011 when the infant vaccination schedule was “high” the IMR had dropped like a rock to ~6.
This fact is the exact opposite to what this idiotic hypothesis predicts.
See this post:

These papers should be retracted for extreme stupidity and incompetence by the authors and peer reviewers … and journal editors.


Beth, thank you for demonstrating how a ‘paper’ like this can cause damage.
Parents want to “assessing risk for their child”, and the authors play on this fear.
As explained above, any claim that the authors research is ‘legitimate’ is severely problematic, to say the least. And any claim that the authors desire to “understand’ anything is not supported by their work or history.
And while their research has been debunked, their actually goal, to plant seeds of doubt, seems to have faired better…

“What is needed to debunk this paper is not a point-by-point analysis of every flaw in it”

Odd. I thought that was an excellent way to debunk not only this bad paper, but deeply flawed research in general.

“The (sic) is information parents want to know with regard to assessing risk for their child.”

No, it’s what antivaxers lust for in vain, while deliberately ignoring all we’ve learned about immune system function.

“the results should be treated as a legitimate signal”

About as valid a “signal” as the ones Neil Z. Miller gets from outer space.

Maybe Beth’s right – those who find this ‘paper’ to be ‘legitimate’, are probably not influenced by factors such as data, research methodology, the skill or history of the authors etc., so “a point-by-point analysis of every flaw in it” is probably a a waste of time.

For them, probably yes. For those on the fence who haven’t gone fully down the rabbit hole, I think there is definite value.

Try as I might, when it comes to searching for and then reading ‘studies’ I am always none the wiser. Most (if not all) of the stuff goes completely over my at supersonic speed. There was a period when I started consulting ‘Dr Google’ and found I had every cancer possible along with various other diseases. For non medical proffesionals like myself, just as others do, rely on those ‘studies’ being broken down and explained with simpler language (we IT support guys called it ‘dumbing down’). But the most sensible thing to do is ask your own doctor.

For me it’s a matter of trust. I trust the medical profession, and I can give a multitude of explanations as to why that comes from my experiences throughout my 64 years. If I did not, I wouldn’t be here now.

OT: In a quasi-GBD-ish column in the NYT today, Bret Stevens claims a Cochrane study definitively proves masks don’t work condemns the CDC for instituting mask mandates, and opines that Dr. Walensky should resign for her grievous error. Happily though, the Times did attach a comment thread, which is filled by notes from people who actually read the study pointing out that it doesn’t say what Stevens says it does at all, and the methodology has some issues as well. One comment likened his version of the study to Bill Barr’s version of the Mueller report.

Stevens is a “sane” conservative, not a gonzo Trumper, and I take this column as evidence that backlash against public health efforts in the pandemic is now “mainstream” GOP orthodoxy.

This is fraught. We count on Cochrane Reviews to help guide practice. The studies they had available for review were not the best quality but they were able to reach a conclusion. I’m not prepared to dismiss it. Our own internal employee health/infection prevention surveys found the same thing-hand washing matters big time masks are neutral at best. This is a tough situation.

Another factor: We have had a lot of staff callouts with URIs, all viral, despite switching back to all staff in N95s a month ago. They are probably picking these bugs up elsewhere but the masks are now universally seen as punishment. Staff see them as something the C-suite can “force” on us.

I doubt I have to spell out where I’m afraid this is going…

And now many in the anti-mask camp are framing this as the NYT ‘admitting’ that masks do nothing. While the review may be problematic, the amplification of it’s conclusions through our own biases is ultimately the biggest problem.

That might be the final straw that gets me to address Tom Jefferson and that Cochrane review on my other blog. Hmmm.

One of the problems when addressing an anti-vax and/ or vaccine hesitant audience is that they probably have already been fed pre-emptive misinformation in regard to SB research and altie self-aggrandisement. Some examples:
— they present anti-vaxxers as experts, leaders in their fields, authors of many studies, professors, well-known authorities.
This is especially common with the current crop of Covid/ PH denialists ( Malone, McCollough, Kory, GBD, FLCCC)
— they present SBM as the “Orthodoxy”- behind the times, corrupted by pharma/ government, money hungry, evil personified
People who follow woo have been “educated” by their leaders over time, shutting out other sources which are labeled false, corrupt, compromised, harmful, inhumane.

Also, their audiences are usually not well informed about how research proceeds- they don’t understand consensus, study design, ethical issues, how statistical analysis is interpreted, how complex virology and vaccine development are

Right now, anti-vaxxers toss out single studies or a single datum as if they meant much!

In addition, people may be attracted to woo/ contrarianism because of personality or social factors. Being a dissenter is their chosen profession; they want to be amongst the first wave of the latest trend of BS.

Working around these issues presents sceptics with a complex, long term task. Some are indeed unreachable. A question I’d ask to differentiate the unreachable is whether they think that their pet hypothesis is being taught in universities: the truly entranced will probably also dismiss universities.

Even in the United States, about 2-4% of children never received any vaccines. That is hundreds of thousands of children.

Many of them have electronic health care records.

It should be straightforward to put together a matched comparison group and see just how much better the vaccinated children are doing, health-wise.

I do realize that there is also a “public benefit” from total vaccination, however such a comparison would be most beneficial.

I am very surprised that such things are not undertaken, to the best of my knowledge.

It should be straightforward to put together a matched comparison group and see just how much better the vaccinated children are doing, health-wise.

This sentence, more than any recent statement from you, tells me just how little you know about epidemiology and epidemiological research. You don’t even know what you don’t know.

It should be straightforward to put together a matched comparison group and see just how much better the vaccinated children are doing, health-wise.

Is it Anne Dachel Appreciation Day or something? Sheesh.

Holy blast from the past!

(And others Idk)

Here may be a nugget for Igor or others: from 08’ Prometheus via Pd in the comments section.

But read on (in the comments from Narad’s link) Prometheus does have a response to why this wouldn’t work after he suggested it.

“This is a study that could be done rather quickly and with a minimum of expense. It would eliminate many of the sources of bias and would fairly easily generate balanced study populations that would be a good match to most of the general population.

[a] Contact a large HMO with actual facilities (e.g. Humana or Kaiser) and arrange to get access to their patient medical records. This is routinely done, although the HMO will want assurances that patient confidentiality will be maintained.

[b] Obtain a list of patients with autism diagnosis in the proper age range (I would suggest 6 – 12 years).

[c] Select one thousand of these patients at random. This would allow you to detect a difference if the prevalence of unvaccinated children is less than 1/3 that in the general population (alpha error level 5%, beta error level 5%). If the difference is less than that, you’ll need to select more subjects.

[d] Confirm the diagnosis by having a child psychiatrist or psychologist review the records.

[e] For each of the remaining children, select a non-autistic control child from the HMO database that is of the same age, sex, geographical region, etc.

[f] Determine how many of the children in each group have received all, none or some of their vaccinations (keep track of which vaccines, when, etc.). If the GR “survey” was right (a very big “if”) about the number of children unvaccinated, each group should have around 30 unvaccinated children, unless there is a correlation between vaccination and autism.

[g] If the autism and non-autism groups have statistically significant differences in their vaccination rates, then a correlation can be claimed. If the study shows no correlation, then the relative risk is less than 3. You’d have to have twice as many subjects to bring the minimal relative risk to below 2.

Using HMO patients eliminates any issues of affordability (which is minimal) or access to health care. Although the population of people who have HMO coverage is not necessarily the same as the overal US population, using the case control design ensures that the two groups are as similar as possible.”

Narad = “agent provocateur”

Sorry Denice I had to…

So that link also led me to read some old comments that I got chastised for… Somehow I stumbled upon the time that I was attempting to explain Statistics…

Can idw56old (the resident phd in statistics) audit this post of mine from 2014? 🙂 it starts with my address to Antaeus below.

Narad, you even apologized at the end to me… And I never said thank you. So, Thank you, Sir, for that.

Squirrelite was thoughtful receptive and kind as always. 🙂 🙂

It was a response to Antaeus Feldspar ( also a player from the past)

Antaeus—“Okay, my brain’s a bit too tired from dealing with the Dilbertia of my day job, so I’m trying to articulate exactly why multiplying P-values is so absurd, and I keep thinking I have it and then it slips away. I have no doubt that it’s wrong, I just can’t articulate why. If you get a result from one study that only had 1-in-20 odds of happening that way by chance, and get a similar 1-in-20 result from a second study, isn’t it correct to say that the odds of getting those results by chance is 1-in-400? I know it must be wrong, but I can’t articulate it, and it’s bugging me.”
Good Morning Antaeus,

I see that no one took the time to work through this with you. I will try (mainly because it is interesting to me, your humility is very much more inviting and becoming of you, and I think you will find interest in it as you seem to be a math/logic whiz) to work through it step by step and hopefully I can explain it clearly enough that you will understand the concept. My only formal education in statistics is one upper level Biostatistics course as an undergrad (I do not have any further degrees.) So, I have a good groundwork of understanding from this class, but I had to read a number of papers to get a good grasp on this (which I will reference throughout and list at the end)

First, let’s start with hypothesis testing. When we test a hypothesis it is customary to establish a null hypothesis (H0) that states something to the effect of “there is no difference between A and B” with the alternative hypothesis (H1) then being “there is a difference between A and B” (this being a two tailed hypothesis, because we are not concerned with the difference being in the positive or negative direction). We then set and alpha level (customarily this is set at .05), any p-value above this level leads us to “fail to reject” the null hypothesis, while a p-value below this level allows us to say that we can reject the null hypothesis (note: these are not the same as saying “it proves” or the alternative hypothesis is correct or anything of this nature, just simply fail to reject and reject.)

The p-value specifically means:

“it is the probability, under the null hypothesis, of obtaining the observed result or any more extreme result” (1)

Also, quickly, when one states that the 95% CI is x-y that doesn’t mean what I read in a comment above (and this is probably the most common misconception in all of statistics). In reality, this means that if I do 100 studies on a parameter/statistic of interest, then in 95 out of 100 of those studies the parameter/statistic will fall with in this interval (sorry for being a statistic pedant, but my biostat prof would be proud of me, he really ingrained this one in our heads.) There are some good papers on CI and p-value and misconceptions, etc. (2,3)

Ok, now that we got the basics out of the way (bear with me if this was a review) let’s move on to what you are really interested in and that is how to properly combine p-values to look at the significance of a number of independent tests taken together (one form of meta-analysis.)

On the assumption that the null hypothesis is true, this sampling distribution is a probability density function with uniform distribution on the interval from 0 to 1. (4) That means that we would expect that these independent p-values would take on a normal distribution.

And this is where you should make the connection. It is not the probability of the intersection of two independent probabilities p1 and p2 (if there are only two p-values under consideration,) but rather it is the probability of obtaining p1 and p2 or any more extreme pair of p-values. (1)

(Reference 1 really does the best job of explaining this concept… I would highly recommend reading it, I bet you will enjoy it.)

The most common way to achieve this is to use Fisher’s Method (but not the only way!)

So the presumption is that the independent P-value (P) is a random variable with a uniform distribution on [0,1]. Then we can say that Y= -ln (P) is a chi-squared with one degree of freedom.(4)

So, if there are k number of independent tests with associated p-values (P1, P2, P3, P4, … Pk) then Yi=-ln(Pi) and we get a list of independently identically distributed random variables that have a chi-square distribution with one degree of freedom (Y1, Y2, Y3, Y4… Yk).(4)

Then we need to sum these i.i.d. variables and we get W= summation symbol (Yi). W being a chi-square distribution with k degrees of freedom.(4)

Then to the final step which brings it all together… we want the area under the chi-squared curve to the right of W (this is the overall p-value) so we can use the chi-square table for this, knowing W and the degrees of freedom.
Bingo!! You got it I hope.

Let’s do a hypothetical example and then discuss some caveats.

I haven’t looked at anything Clarke actually wrote, but let’s say he was trying to combine 5 p-values and the values were:

.341, .116, .543, .222, .078. Note that there are no significant p-values in this list and finding the intersection of these we come to a p-value of .00371 (highly significant… or is it??)

Ok, now let us apply what I just explained

-ln(.341)=1.0758, -ln(.116)=2.154, -ln(.543)=.6106, -ln(.222)=1.505, -ln(.078)=2.55

So, W=1.0758+2.154+.6106+1.505+2.55=7.8956

And we have 5 degrees of freedom. Now let’s use the chi-squared table or calculator and plug in both the dof and W. We get an overall p-value of .1621 (not even close to .00371)

Some caveats:

This is not the only method to combine p-values.
This is an un-weighted method to combine p-values.

I also found this source (5) that was a really brief but concise explanation (I have a feeling that you will also find this interesting) on combining p-values with a different formula, written out below (god damnit, I wish I was better at HTML and could write this how it should look)

K= product of all of the independent p-values (P1P2P3P4…Pn)

n= number of independent p-values

K * summation symbol, where i=0 is under the summation symbol and n-1 is above summation symbol and to the right of the summation symbol is ((-ln(k))^i)/i!

(can somebody (narad) please apply some HTML knowledge to this so it looks like it should, thanks.)

For a discussion on caveat 2, see source (6) as it discusses the z-test and weighted z-test in comparison to Fisher’s method and how they are superior to Fisher’s method.

Hope this helps!

Elston R.C. “On Fisher’s Method of Combining p-Values” Biom. J. 33 (1991) 3, 339-345
PMC2689604 (sorry, Im being lazy)
Whitlock M.C. “Combining probability from independent tests: the weighted Z-Method is superior to Fisher’s approach” Journal of Evolutionary Biology. 18 (2005) 5, 1368-1373 (more discussion on combining p-values)

can somebody (narad) please apply some HTML knowledge to this so it looks like it should, thanks.

I missed this comment earlier. If you’re familiar with LaTeX, WordPress <a href=”>supports it. In some cases, one can get away with Unicode, but it doesn’t even have a full set of superscripts and subscripts. So,




Erm, that didn’t come out quite right, but I think you’ll get the idea.

For England, it is indeed easy to make statistics abou deah by vaccination status. ONS has a new dataset:
Check it. Now whole 2022 is covered
You may (dis)like German data by KiGGS, too:
“Children with a sufficient TDPHiHeP vaccination at the end of the 1st year of life had a lower risk of being diagnosed with hay fever after the 1st year of life (adjusted prevalence ratio 0.85, 95% confidence interval 0.76–0.96). Analyses for associations between TDPHiHeP vaccination and risk of atopic dermatitis or asthma, or between age at onset of vaccination or of the number of antigens vaccinated in the 1st year of life and risk of atopic disease failed to yield statistical significance.”

Should I tell Igor why this study would be difficult?
At any rate, he can look at JAMA, April 2015, Anjali Jain et al, who elucidate the issue presenting a viable solution and realistic answer: no association between MMR and ASD dx.
Also, KiGGS, quoted by Aarno.

Comments are closed.


Subscribe now to keep reading and get access to the full archive.

Continue reading