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From a news report:

Khandaker and colleagues found that participants who were exposed to infections, especially in early childhood, were more likely to have lower IQ (adjusted mean difference for infection at birth to age 1 year = –1.61; 95% CI, –1.74 to –1.47) and increased risk for nonaffective psychosis in adulthood (adjusted HR = 1.19; 95% CI, 1.06-1.33).

What is "adjusted mean difference" in plain English? I found that "mean difference" can have two meanings, but either is hard to understand. And what is adjusted in this case? How was it adjusted? And what is "difference"? What was subtracted from what?

I looked up the original publication in JAMA (free full text):

Cox regression was used to calculate the hazard ratio (HR) and 95% confidence interval for NAP (non-affective psychosis) for those exposed to childhood infection compared with those who were unexposed. Linear regression was used to calculate the mean difference in IQ between those exposed to childhood infection compared with unexposed.

Did they first calculate the mean IQ for the group with infections, then the mean IQ for the group without infection, and then subtracted one figure from the other? And finally, they adjusted this difference? Or did they first adjust (I don't know how) each of the two figures, before subtraction?

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    $\begingroup$ "Adjusted" implies they controlled for covariates, so it likely wasn't as simple as subtracting one group estimate from another. Your first clue is the sentence beginning "The Medical Birth Registry and the Population and Housing Census data were used for birth-related and sociodemographic data..." The existence of a section called "Adjustment for Covariates" confirms what I have surmised. More information is available in the "Statistical Analysis" section. $\endgroup$
    – whuber
    Commented Feb 20, 2018 at 5:00
  • $\begingroup$ @whuber - were numerous pairs of figures used in subtraction? Or was it a single subtraction in the end? (I've just read the Statistical Analysis section and understood several words here and there). Were numerous subtractions made, and then a mean calculated for the obtained differences? If yes, how did they pair persons with and without infection? $\endgroup$ Commented Feb 20, 2018 at 5:10

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When we talk about the term "adjusted" in the context of statistical findings, it indicates that the researchers have taken into consideration other influencing variables or covariates. In simple words, they've made allowances for other factors which might otherwise distort the results. This ensures that the differences we see are genuinely because of the primary variable under investigation—in this case, infections during early childhood—and not because of some unrelated factors.

Your reference to "The Medical Birth Registry and the Population and Housing Census data were used for birth-related and socio-demographic data..." is quite telling. These datasets would have provided additional parameters that the researchers would have wanted to keep constant, like socio-demographic details that could affect both IQ and the chances of having infections. The existence of an "Adjustment for Covariates" section confirms this understanding.

So, when they mention "adjusted mean difference", it's not just a simple matter of subtracting the average IQ of one group from the other. What they've likely done is employed a statistical method (like linear regression) to gauge the difference between these groups while simultaneously accounting for other influencing factors. In layman's terms, they're trying to find out, "After ensuring other external factors are not muddying the waters, what's the actual difference in IQ between children who had infections and those who didn't?"

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