What is Bartlett's theory?

Question

This neuroimaging paper has been cited thousands of times. In it, a method is proposed for computing the correlations among several seed regions and all other brain voxels. Part of this method involves z-scoring Fisher z values:

To combine results across subjects and compute statistical significance, correlation coefficients were converted to a normal distribution by Fischer's [sic] z transform (25). These values were converted to z scores (i.e., zero mean, unit variance, Gaussian distributions) by dividing by the square root of the variance, computed as $1 / \sqrt{(n - 3)}$, where $n$ is the degrees of freedom in the measurement. Because individual time points in the BOLD signal are not statistically independent, the degrees of freedom must be corrected according to Bartlett's theory (25). The correction factor for independent frames was calculated to be 2.34, resulting in 318 / 2.34 = 135.9 df.

No other mention is made of Bartlett's theory -- that is, they don't explain how they came to the value of 2.34.

Reference 25 is this textbook. Although the textbook seems excellent and reasonably priced, I would like to acquire a basic understanding of Bartlett's theory without having to buy it, so that I might replicate this method. Of course my first course of action was to perform several Google searches; they, however, turned up nothing.

What is Bartlett's theory, in this context?

Answer 1

I suspect what Fox et al 2005 refers to is Bartlett 1946 which is a more "general" form of the AR1-based variance estimator (Bartlett 1935). Bartlett 1946's estimator was later adapted for bivariate time series by Quenouille 1947 as a DoF estimator.

Suppose $X$ and $Y$ are two time series of length $N$ where $\rho_{XX,k}$ and $\rho_{YY,k}$ are the autocorrelation coefficients of $X$ and $Y$, respectively, on lag $k$. Then Quenouille 1947 found the effective DoF to be

$$ \hat{N} = N \left(\sum_{k=-\infty}^{\infty} {\rho}_{XX,k} {\rho}_{YY,k}\right)^{-1}, $$

while Bayley & Hammersley 1946 found, $$ \hat{N} = N\Big(1+2\sum_{k=1}^{N-1}\frac{(N-k)}{N}\rho_{XX,k}{\rho}_{YY,k}\Big)^{-1}. $$

There are many approximations of Bartlett's original estimator. One nice review of these variants can be found in Pyper and Peterman 1998.

It is however very important to note that all above estimators assume $X$ and $Y$ are uncorrelated ($\rho = 0$, which in neuroimaging is far from reality). The problem is that once the assumption is violated, these estimators remarkably overestimate the variance due to a confounding of autocorrelation and crosscorrelation, a phenomena also known as statistical aliasing; see Appendix D of Afyouni et al 2018.

So: No correction over-estimates DoF (underestimates variance) and the above corrections under-estimates DoF (overestimates variance). What can be done? See the estimator has recently been proposed in Afyouni et al 2018, \begin{equation} \begin{split} \mathbb{V}({\hat\rho})&=N^{-2}\left[\vphantom{\sum_k^M}(N-1)(1-\rho^2)^2 \right. \\ &\quad +\rho^2 \sum_k^M w_k (\rho_{XX,k}^2 + \rho_{YY,k}^2 + \rho_{XY,k}^2 + \rho_{XY,-k}^2)\\ &\quad -2 \rho \sum_k^M w_k (\rho_{XX,k} + \rho_{YY,k}) (\rho_{XY,k} + \rho_{XY,-k}) \\ &\quad +2 \left.\sum_k^M w_k (\rho_{XX,k}\rho_{YY,k}+\rho_{XY,k}\rho_{XY,-k}) \right], \end{split} \label{Eq:fastMEIntro} \end{equation}

where $w_i=N-2-k$. While this is an involved expression, we show that -- with sensible regularisation of the autocorrelation and crosscorrelation function -- this gives accurate DoF / variance estimates over a range of settings. (See also Roy 1989 for an asymptotic derivation of the same).

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Bartlett, M. S. (1946). On the Theoretical Specification and Sampling Properties of Autocorrelated Time-Series. Supplement to the Journal of the Royal Statistical Society, 8(1), 27. http://doi.org/10.2307/2983611

Bartlett, M. S. (1935). Some Aspects of the Time-Correlation Problem in Regard to Tests of Significance. Journal of the Royal Statistical Society, 98(3), 536. http://doi.org/10.2307/2342284

Quenouille, M. H. (1947). Notes on the Calculation of Autocorrelations of Linear Autoregressive Schemes. Biometrika, 34(3/4), 365. http://doi.org/10.2307/2332450

Bayley, G. V., & Hammersley, J. M. (1946). The “Effective” Number of Independent Observations in an Autocorrelated Time Series. Supplement to the Journal of the Royal Statistical Society, 8(2), 184. http://doi.org/10.2307/2983560

Pyper, B. J., & Peterman, R. M. (1998). Comparison of methods to account for autocorrelation in correlation analyses of fish data, 2140, 2127–2140.

Afyouni, Soroosh, Stephen M. Smith, and Thomas E. Nichols. "Effective Degrees of Freedom of the Pearson's Correlation Coefficient under Serial Correlation." bioRxiv (2018): 453795. https://www.biorxiv.org/content/early/2018/10/25/453795

Roy, R. (1989). Asymptotic covariance structure of serial correlations in multivariate time series. Biometrika, 76(4), 824–827. http://doi.org/10.1093/biomet/76.4.824

Answer 2

In the supporting information for another paper from the same group, I found this elaboration:

Because individual time points in the BOLD signal are not statistically independent, the degrees of freedom must be computed according to Bartlett’s theory, i.e., computing the integral across all time of the square of the autocorrelation function (11).

Reference 11 is the same Jenkins and Watts textbook.

This raises the question, Which autocorrelation function is used: the autocorrelation for the seed region or the autocorrelation for the voxel with which it has been correlated? And what is meant by "across all time" -- across all time lags?

Michael Fox, in a personal communication, responds that the autocorrelation function for the seed region is used, and that "across all time" indeed means across all time lags.

(But why use the autocorrelation for the seed region and ignore the other timeseries?)

I can't find any independent support for this approach.

A handout I found from a course in Atmospheric Sciences at U Dub suggests a vaguely similar approach and attributes it to Bartlett:

Indeed, Bretherton et al, (1999) show that, assuming that one is looking at quadratic statistics, such as variance and covariance analysis between two variables $x_1$ and $x_2$, and using Gaussian red noise as a model then a good approximation to use is:

$\frac{N^*}{N} = \frac{1− r_1(\Delta t)r_2 (\Delta t)}{ 1+ r_1(\Delta t)r_2 (\Delta t)}$

where, of course, if we are covarying a variable with itself, $r_1(\Delta t)r_2(\Delta t) = r(\Delta t)^2$. This goes back as far as Bartlett (1935). Of course, if the time or space series is not Gaussian red noise, then the formula is not accurate. But it is still good practice to use it.

In this, $N^*$ is the degrees of freedom, $N$ is the number of observations, and $r_x$ is the autocorrelation function for signal $x$.

The operative words here are "vaguely similar."

I have now read a good portion of Jenkins and Watts, and skimmed through even more, but have found no description in the text of the correction described by Fox et al. or of the Fisher z-transform.

All that said, I have decided to compute the correction factor the same way Michael says I should . . . but wait: How do I do this?

Answer 3

Bartlett's theory here refers to results from this paper On the Theoretical Specification and Sampling Properties of Autocorrelated Time-Series.

The main idea here is that if you have $n$ discrete time observations, the effective number of degrees of freedom needs to be less than n since the observations are not independent, given by $n \cdot\text{correction factor}$

For a linear autoregressive model with $s$ lags
$$ x_{t+1} = \sum_{l=1}^{s}x_{t-l}\rho_{l} + e_t $$

Suppose $s=1$. The asymptotic correction factor for AR-(1) model is given by $\frac{1-\rho^2}{1+\rho^2}$. Thus instead of an asymptotic variance of $\frac{1}{n}$, we have a larger asymptotic variance of $\frac{1}{n}\frac{1+\rho^2}{1-\rho^2}$

You can similarly find a derivation of Bartlett's formula for asymptotic variance vector of $(\rho_1,\ldots, \rho_s)$ autocorrelation coefficients case in Theorem 7.2.1 in [Brockwell and Davis][2] and for a number of other time-series models as well.

For the case of asymptotic variance between two different time-series, $x(t)$ and $y(t)$, a similar formula applies and is nicely discussed in this paper by [Haugh][3]. Even if you do whiten each individual time-series first, the test statistic for the sample cross-correlation still needs to adjust the asymptotic variance estimate.

[2]: Time Series: Theory and Methods by Brockwell and Davis. Available for free at http://link.springer.com/book/10.1007%2F978-1-4419-0320-4

[3]: Haugh, Larry D. "Checking the independence of two covariance-stationary time series: a univariate residual cross-correlation approach." Journal of the American Statistical Association 71.354 (1976): 378-385. APA