Here we are going to see how to combine polls from different sources.

Let us consider again Nevada polls.

Poll | Date | Sample | MoE | Clinton (D) | Trump (R) | Johnson (L) | Spread | |
---|---|---|---|---|---|---|---|---|

0 | RCP Average | 7/7 - 8/5 | -- | -- | 43 | 40.7 | 6.3 | Clinton +2.3 |

1 | CBS News/YouGov* | 8/2 - 8/5 | 993 LV | 4.6 | 43 | 41.0 | 4.0 | Clinton +2 |

2 | KTNV/Rasmussen | 7/29 - 7/31 | 750 LV | 4.0 | 41 | 40.0 | 10.0 | Clinton +1 |

3 | Monmouth | 7/7 - 7/10 | 408 LV | 4.9 | 45 | 41.0 | 5.0 | Clinton +4 |

Instead of doing an average of the poll as it is done by RCP (RealClearPolitics), we use Covariance Intersection.

**Covariance intersection**is an algorithm for combining two or more data source when the correlation between them is unknown.

Let us denote with \(\hat{a}\) a vector of observations (e.g., 43,41,16 from CBS News/YouGov) and \(\hat{b}\) another vector of observations (e.g., 41,40,19 from KTNV/Rasmussen). \(A\) denotes the reliability of the data poll \(\hat{a}\) that we assume to be equal \(1/sample size\) (e.g., 1/993

for CBS News/YouGov) and \(B\) denotes the reliability of the data poll \(\hat{b}\) (e.g., 1/750 for KTNV/Rasmussen).

Given the weight \(\omega\),Covariance Intersection provides a formula to combine them:

$$ C^{{-1}}=\omega A^{{-1}}+(1-\omega )B^{{-1}}\,, $$ $$ \hat{c} =C(\omega A^{{-1}}{\hat a}+(1-\omega )B^{{-1}}{\hat b})\,. $$

This formula can be extended to an arbitrary number of sources. For instance, for the previous table using uniform weights \(\omega_1=1/3,\omega_3=1/3,\omega_3=1/3\), we get

$$ C^{{-1}}=\omega_1 993+\omega_2 750+\omega_3 408=717$$ $$ \hat{c} =C(\omega_1 993{[43,41,16]}+\omega_2 750[41,40,19]+\omega_3 408[45,41,14])\,. $$

The final result is

$$

C^{{-1}}=717, ~~~\hat{c}=[42.68, 40.65, 16.67]

$$

It can be observed that by using \(\omega_1=1/3,\omega_3=1/3,\omega_3=1/3\) the combined poll \(\hat{c}\) reduces to the average of the input polls

weighted by the sample size. However, it is possible to choose other values of the weights, see for instance here.

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