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I think this work would be better as a short communication, rather than a full paper - given its cross-sectional nature and the fact that it confirms previous findings but does not present anything particuarly new. The value of it, is that it describes this association in a population where it hasn't been studied before and thus adds to the general knowledge of population health in the region.
I agree with the recommendation of including interpregnancy period and occupation as possible confounders and with the suggestion of having a short communication.
It would be more appropriate to report SE instead of SD since you are analyzing representative data of a DHS from Peru and methods section inform svy commands were used so I believe you have SE instead.
Another issue is how exclusions were handled. When analyzing complex survey data, there is a strong recommendation to not exclude cases but to work with subpopulations in an unconditional analysis in order to guarantee correct variance estimation. How was the exclusions performed? Cases were deleted?
You claim that your results demonstrates that place of residence and wealth index affect the relationship between parity and overweight or obesity, however your data demonstrates the opposite in Tables 4 and 5. In all your stratified analyses (crude and adjusted), the 95%CI of each comparable parity category overlaps for each strata. For instance, for women with 2 children the 95%CIs of the crude PR value for obesity are 4.0-9.3, 4.5-8.4, 3.8-6.8, and 3.8-6.4 for each wealth index, respectively. This is true for every parity or place of residency category and overweight and its subtypes. This suggest that in your sample neither wealth index nor place of residence alters the relation between parity and overweight.
Obesity (BMI>30) is a subtype of overweight (BMI>25). As shorthand, overweight but not obese (BMI>25 but <=30) is referred also as overweight. There are not three types of “overnutrition” as you claim in your manuscript. There is overweight, divided in two subtypes, obesity and overweight (but not obese). That is overweight=overweight (but not obese)+obesity. In both, tables 4 and 5, you have that overweight (but not obese)+obesity does not equals overweight. So this could be a typo in numbers, but if not, then your results are incorrect.
In my opinion, overnutrition is a vague and incorrect term. It would be odd to consider that overweight and/or obese individuals have too much nutrition. By the same logic, some types of hyperthyroidism or hypervitaminosis can also be overnutrition, but one would not use this term for these conditions. However, if we were to use them, then overnutrition would be a non-specific term that would be incorrectly used, exchangebly, to refer specifically to overweight or obesity as you are doing in your manuscript. I think it is better to always be specific when refering to overweight and/or obesity.