$$TPP - Coalition_p = \beta_1 Greens_p + \beta_2 OneNation_p + \beta_3 Other_p + \epsilon$$
Which, in matrix notation, we will simplify as:
$$ y = X \beta + \epsilon $$
In this simplification, \(y\) is a column vector of (TPP - Coalition primary) vote estimates from a pollster. \(X\) is the regression design matrix, with k columns (one for each party's primary vote) and N rows (one for each of the reported poll results). \(\beta\) is a column vector of party coefficients we are seeking to find through the regression process. And \(\epsilon\) is a column vector of error terms which we assume are independent and identically distributed (iid) with a mean of \(0\). Through the magic of mathematics we can seek to minimize the sum of the squared errors using algebra and calculus to show:
$$ \sum_{i=1}^n\epsilon_i^2 = \epsilon'\epsilon = (y-X\beta)'(y-X\beta) $$
$$= y'y - \beta'X'y - y'X\beta + \beta'X'X\beta $$
$$= y'y - 2\beta'X'y + \beta'X'X\beta $$
From this last equation, we can use calculus to find the \(\beta\) that minimizes the sum of the errors squared:
$$\frac{\partial \epsilon'\epsilon}{\partial\beta} = -2X'y+2X'X\beta = 0$$
Which can be re-arranged to the famous "ex prime ex inverse ex prime why":
$$ \beta = (X'X)^{-1}X'y $$
Before I get to the results, there are a few caveats to go through. First, not all of the primary vote poll data sums to 100 per cent. In the data I looked at, the following polls did not sum to 100 per cent.
--- Does not add to 100% ---
L/NP ALP GRN ONP OTH Sum Firm Date
12 35.0 38.0 10.0 7.0 9.0 99.0 Essential 12 Dec 2017
18 31.0 34.0 11.0 11.0 14.0 101.0 YouGov 14 Nov 2017
19 36.0 38.0 9.0 8.0 10.0 101.0 Essential 14 Nov 2017
21 36.0 37.0 10.0 7.0 9.0 99.0 Essential 30 Oct 2017
25 36.0 38.0 10.0 7.0 10.0 101.0 Essential 4 Oct 2017
32 35.0 34.0 14.0 1.0 15.0 99.0 Ipsos 6-9 Sep 2017
63 36.0 37.0 10.0 8.0 10.0 101.0 Essential 13-16 Apr 2017
67 35.0 37.0 10.0 8.0 11.0 101.0 Essential 24-27 Mar 2017
69 34.0 37.0 9.0 10.0 9.0 99.0 Essential 17-20 Mar 2017
75 36.0 35.0 9.0 10.0 9.0 99.0 Essential 9-12 Feb 2017
77 35.0 37.0 10.0 9.0 8.0 99.0 Essential 20-23 Jan 2017
80 37.0 37.0 9.0 7.0 9.0 99.0 Essential 9-12 Dec 2016
83 36.0 30.0 16.0 7.0 9.0 98.0 Ipsos 24-26 Nov 2016
88 37.0 37.0 11.0 5.0 9.0 99.0 Essential 14-17 Oct 2016
92 38.0 37.0 10.0 5.0 11.0 101.0 Essential 9-12 Sep 2016
109 34.0 35.0 11.0 8.0 13.0 101.0 YouGov 7-10 Dec 2017
110 32.0 32.0 10.0 11.0 16.0 101.0 YouGov 23-27 Nov 2017
----------------------------
To manage this, I normalised all of the primary vote poll results so that they summed to 1.
The second thing I did was limit my analysis to those pollsters that had more than 10 polls since the last election. This meant I limited my analysis to polls from Newspoll and Essential.
Let's look at the multiple regression results. The key results is the block in the middle, with the three parties in the left hand column: GRN, ONP and OTH - Greens, One Nation and Others. The first coefficient column is the best linear unbiased estimate of the preference flows to the Coalition from each of these parties. The 95 per cent confidence intervals can be seen in the far right columns.
---- Essential ----
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.999
Model: OLS Adj. R-squared: 0.999
Method: Least Squares F-statistic: 1.173e+04
Date: Sat, 17 Mar 2018 Prob (F-statistic): 2.09e-61
Time: 13:12:35 Log-Likelihood: 190.20
No. Observations: 45 AIC: -374.4
Df Residuals: 42 BIC: -369.0
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
GRN 0.2236 0.054 4.162 0.000 0.115 0.332
ONP 0.4396 0.037 11.775 0.000 0.364 0.515
OTH 0.5091 0.050 10.190 0.000 0.408 0.610
==============================================================================
Omnibus: 3.681 Durbin-Watson: 1.938
Prob(Omnibus): 0.159 Jarque-Bera (JB): 1.691
Skew: -0.020 Prob(JB): 0.429
Kurtosis: 2.051 Cond. No. 20.0
==============================================================================
---- Newspoll ----
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.999
Model: OLS Adj. R-squared: 0.999
Method: Least Squares F-statistic: 7414.
Date: Sat, 17 Mar 2018 Prob (F-statistic): 1.81e-34
Time: 13:12:38 Log-Likelihood: 110.81
No. Observations: 26 AIC: -215.6
Df Residuals: 23 BIC: -211.9
Df Model: 3
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
GRN 0.2581 0.090 2.881 0.008 0.073 0.443
ONP 0.4976 0.035 14.202 0.000 0.425 0.570
OTH 0.4424 0.084 5.237 0.000 0.268 0.617
==============================================================================
Omnibus: 2.886 Durbin-Watson: 0.753
Prob(Omnibus): 0.236 Jarque-Bera (JB): 1.437
Skew: -0.212 Prob(JB): 0.488
Kurtosis: 1.929 Cond. No. 26.8
==============================================================================
From the Ordinary Least Squares (OLS) multiple regression analysis, our best guess is that Essential flows 22 per cent of the Green vote to the Coalition, it flows 44 per cent of the One Nation vote, and it flows 51 per cent of the Other vote. In comparison, Newspoll flows 26 per cent of the Green vote to the Coalition, 50 per cent of the One Nation vote, and 44 per cent of the Other vote.
While we get an estimate of preference flows to the Coalition for each of the three parties from both pollsters, it is worth noting that small sample sizes have resulted in quite wide confidence intervals for these estimates.
We can also do a Bayesian multiple linear regression. The Stan model I used for this follows.
// STAN: multiple regression - no intercept - positive coefficients
data {
// data size
int<lower=1> k; // number of pollster firms
int<lower=k+1> N; // number of polls
vector<lower=0,upper=1>[N] y; // response vector
matrix<lower=0,upper=1>[N, k] X; // design matrix
}
parameters {
vector<lower=0>[k] beta; // positive regression coefficients
real<lower=0> sigma; // standard deviation on iid error term
}
model {
beta ~ normal(0, 0.5); // half normal prior
sigma ~ cauchy(0, 0.01); // half cauchy prior
y ~ normal(X * beta, sigma);// regression model
}
The results I got were very similar to the standard OLS results. In this case I have identified the 95% credible interval, which is the Bayesian equivalent of the confidence interval. Again, the One Nation and Other results are quite different, and swapped about between the two pollsters.
From Stan: for Essential
2.5% median 97.5%
GRN 0.121745 0.228054 0.335852
ONP 0.362631 0.438261 0.513884
OTH 0.404663 0.506283 0.607355
From Stan: for Newspoll
2.5% median 97.5%
GRN 0.089876 0.267659 0.450531
ONP 0.420720 0.494935 0.566663
OTH 0.259106 0.433838 0.601585
From the Bayesian analysis, our best guess is that Essential flows 23 per cent of the Green vote to the Coalition. It flows 44 per cent of the One Nation vote. And it flows 51 per cent of the Other vote. In comparison, Newspoll flows 27 per cent of the Green vote to the Coalition, 49 per cent of the One Nation vote, and 43 per cent of the Other vote.
These Bayesian results can be charted as probability densities as follows. In these charts the median sample for each distribution is highlighted with a thin vertical line.
For completeness, the supporting Python program that generated this analysis follows.
# PYTHON - estimates of preference flows from polling data
import pandas as pd
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
import sys
sys.path.append( '../bin' )
from stan_cache import stan_cache
# --- chart results
graph_dir = './Graphs/'
walk_leader = 'STAN-PREFERENCE-FLOWS-'
plt.style.use('../bin/markgraph.mplstyle')
# --- curate data for analysis
workbook = pd.ExcelFile('./Data/poll-data.xlsx')
df = workbook.parse('Data')
# drop polls without one nation
df = df[df['ONP'].notnull()]
# drop pre-2016 election data
df['MidDate'] = [pd.Period(d, freq='D') for d in df['MidDate']]
df = df[df['MidDate'] > pd.Period('2016-07-04', freq='D')]
# normalise the data - still in 0 to 100 range
parties = ['GRN', 'ONP', 'OTH']
all = ['L/NP', 'ALP'] + parties
df['Sum'] = df[all].sum(axis=1)
bad = df[all + ['Sum', 'Firm', 'Date']]
bad = bad[(bad['Sum'] < 99.5) | (bad['Sum'] > 100.5)]
print('--- Does not add to 100% ---')
print(bad)
print('----------------------------')
df[all] = df[all].div(df[all].sum(axis=1), axis=0) * 100.0
# --- Analyse the curated data
firms = df['Firm'].unique()
for firm in firms:
cases = df[df['Firm']==firm]
if len(cases) <= 10:
continue # not enough to analyse
# --- classic OLS multiple regression
# get response vector and design matrix in 0 to 1 range
y = (cases['TPP L/NP'] - cases['L/NP']) / 100.0
X = cases[parties] / 100.0
# regression estimation
model = sm.OLS(y, X).fit()
print('\n\n---- {} ----'.format(firm))
# Print out the statistics
print(model.summary())
# --- let's do the same thing with Stan
# input data
data = {
'y': y,
'X' : X,
'N': len(y),
'k': len(X.columns)
}
# helpers
quants = [2.5, 50, 97.5]
labels = ['2.5%', 'median', '97.5%']
with open ("./Models/preference flows.stan", "r") as file:
model_code = file.read()
file.close()
# model
stan = stan_cache(model_code=model_code, model_name='preference flows')
fit = stan.sampling(data=data, iter=10000, chains=5)
results = fit.extract()
# capture the coefficients
coefficients = results['beta']
print('--- Coefficient Shape: {} ---'.format(coefficients.shape))
estimates = pd.DataFrame()
for i, party in zip(range(len(parties)), parties):
q = np.percentile(coefficients[:,i], quants)
row = pd.DataFrame(q, index=labels, columns=[party]).T
estimates = estimates.append(row)
# capture sigma
sigma = results['sigma'].T
q = np.percentile(sigma, quants)
row = pd.DataFrame(q, index=labels, columns=['sigma']).T
estimates = estimates.append(row)
# print results from Stan
print('From Stan: for {}'.format(firm))
print(estimates)
# plot results from Stan
coefficients = pd.DataFrame(coefficients, columns=parties)
ax = coefficients.plot.kde(color=['darkgreen', 'goldenrod', 'orchid'])
ax.set_title('Kernel Density for Preference Flows by {}'.format(firm))
ax.set_ylabel('KDE')
ax.set_xlabel('Estimated Flow (Proportion)')
for i in coefficients.columns:
ax.axvline(x=coefficients[i].median(), color='gray', linewidth=0.5)
fig = ax.figure
fig.set_size_inches(8, 4)
fig.tight_layout(pad=1)
fig.text(0.99, 0.01, 'marktheballot.blogspot.com.au',
ha='right', va='bottom', fontsize='x-small',
fontstyle='italic', color='#999999')
fig.savefig(graph_dir+walk_leader+firm+'.png', dpi=125)
plt.close()
No comments:
Post a Comment