<i>kpop</i>: a kernel balancing approach for reducing specification assumptions in survey weighting

With the precipitous decline in response rates, researchers and pollsters have been left with highly nonrepresentative samples, relying on constructed weights to make these samples representative of the desired target population. Though practitioners employ valuable expert knowledge to choose what variables <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi></mml:mrow> </mml:math> must be adjusted for, they rarely defend particular functional forms relating these variables to the response process or the outcome. Unfortunately, commonly used calibration weights-which make the weighted mean of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi></mml:mrow> </mml:math> in the sample equal that of the population-only ensure correct adjustment when the portion of the outcome and the response process left unexplained by linear functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi></mml:mrow> </mml:math> are independent. To alleviate this functional form dependency, we describe kernel balancing for population weighting (<i>kpop</i>). This approach replaces the design matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>X</mml:mtext></mml:mrow> </mml:math> with a kernel matrix, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>K</mml:mtext></mml:mrow> </mml:math> encoding high-order information about <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>X</mml:mtext></mml:mrow> </mml:math> . Weights are then found to make the weighted average row of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>K</mml:mtext></mml:mrow> </mml:math> among sampled units approximately equal to that of the target population. This produces good calibration on a wide range of smooth functions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi></mml:mrow> </mml:math> , without relying on the user to decide which <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi></mml:mrow> </mml:math> or what functions of them to include. We describe the method and illustrate it by application to polling data from the 2016 US presidential election.