# Weighting

Weighting is a technique which adjusts the results of a survey to bring them into line with some known characteristics of the population. For example, if a sample contains 40% males and the population contains 49% males, weighting can be used to correct the data to correct for this discrepancy.

# Worked example of sampling weights

Consider the following data showing 10 people’s favorite celebrity:

Brad Pitt
Tiger Woods
Tiger Woods

In this sample of ten, 80% of people have nominated Brad Pitt as their favorite celebrity. When we conduct market research studies, the whole point is to draw conclusions about the population, rather than just our sample, so we should go one step further and, rather than say “80% of the sample nominate Brad Pitt as their favorite celebrity” we should instead be conducting research so that we can say “Brad Pitt is the most popular celebrity, with 80% of people nominating him as their favorite.

Now consider the impact of some additional information about the gender of our ten respondents:

Brad Pitt		Female
Tiger Woods		Male
Tiger Woods		Male

From this data we can see that the sample is unrepresentative in terms of gender, with eight of ten respondents being female (80%) and two being male (20%), compared to the true representation in the world of about 50% men and 50% women. Furthermore, gender seems to be the sole determinant of preference. As the sample is not representative in terms of gender, and gender is correlated with our measure of favorite celebrity, it follows that any estimate of the favorite celebrity will only be valid if we take into account the over-representation of women in the sample.

We can improve our estimate by weighting. A weight is computed for every respondent in a sample, and it is computed by dividing the correct proportion by the observed proportion. The correct proportion of males in our population is 50% and the observed proportion is 20%, so the weight for each male is 50%/20%=2.5 and the weight for each female is 50%/80%=0.625. Thus, our data becomes:

Favourite
Celebrity      Weight
Tiger Woods	2.5
Tiger Woods	2.5 

We now compute our estimate of the proportion of people to have Brad Pitt as their favorite celebrity by summing up the weights of each of the respondents to prefer Brad Pitt and dividing this by the sum of all of the respondents’ weights:

$\frac{8 \times 0.625}{2 \times 2.5 + 8 \times 0.625} = \frac{5}{10} = 50\%$


The approach described here for computing a weight is a relatively simple case, but the basic idea can be extended to deal with much more complicated cases (e.g., it is routine to simultaneously weight by geography, gender and age).

# How weighting works in practice

1. A weight variable is created, by either target weighting or rim weighting.
2. This weight variable is included in analyses, using methods specifically designed to address the weights.

# The impact of weights on statistical inference

Special-purpose statistical methods are required for sampling weights. That is, the standard statistical tests that are described in introductory textbooks are not applicable (in particular, the assumption that the sample has been drawn independently and identically from a statistical distribution is violated). Weight Calibration is commonly used to analyze sampling weights in software that that can only accommodate frequency weights.

# When sample weighting does and does not work

Consider the situation where a survey was well managed but due to some anticipated quirk of how the data was collected it over-represent males. If gender is known to relate to other variables of interest in the survey then it follows both that:

• The data needs to be weighted or otherwise any analyses will be misleading (as they will be biased by the misrepresentation of gender).
• The weighting will fix the data.

However, what if the gender problem is one of many unknown problems in the data. Imagine that the survey is also biased towards low income people, towards people that like to do surveys and towards people that do not get out of doors much. Fixing the gender problem by weighting will not fix any of these other problems. Furthermore, in general we will not know that we have these other problems, as we will have no data to check for representativeness.

The distinction between these two cases is crucial. Where the data is observed to be unrepresentative, weighting will only fix the data if there is good reason to believe that this is the only way in which the data will be representative.