In my most recent post, I discussed hypothetical weather trends over the upcoming 5, 8 or 10 years that could, falsify the consensus position of climate chance, should they occur. In this post, I’ll discuss how I came up with the numbers statistics.

The steps to coming up with numerical values require me to:

1. Identify a hypothesis which I will treat is “the null hypothesis”. That means, that from a statistical point of view, I will consider it provisionally true unless falsified. This is the predicted trend.
2. Select an analytical method and a data set on with to base a hypothesis test: I choose OLS and GISS Land/Ocean data.
3. Estimate standard errors associated with OLS to weather trends from historical data.
4. Find the intervals that would falsify the hypothesis at the 95% confidence interval.

The Hypothesis.

The IPCC document project a range of possible trends for surface temperature increase over the next century. They also show how project that the trend should vary in time. However, over the next two decades, the climate models predict that Global Mean Surface Temperatures are expected to rise at a rate of m_p= +2C/century over the next two decades. (Here, “m” will always imply a trend in degrees/time and the subscript “p” means projected.)

Since the IPCC made this projection, we’d like to compare it to upcoming weather and see if it appears to be correct. For the purpose of this exercise, we will assume the IPCC projection is provisionally correct. That is: we will treat it as the null hypothesis in a statistical test and only reject it if we can show it is inconsistent with data we collect from 2008-2017 inclusive. More specifically, I will reject the IPCC projection only if the temperature trend is too low to be consistent with the IPCC projection. If the trend is higher than projected, I will consider the IPCC projection confirmed.

So, my “null” hypothesis, H_o for the purpose of testing is:

(1)H_o: m_p =+2C/century ≤ m

where “m” is the true, and always unknowable, value.
When we run experiments, of any sort, due to the random nature of weather, our result will provide an estimate of ‘m’, which we will call the sample slope, “m_s“. This quantity will be assumed to be normally distributed around the true mean “m”, with a standard deviation σ_m.

The figure to the left shows the distribution of m_s one might expect if the mean really were 2C/century and the standard distribution for experiments was were 1.1C/century. (This standard distribution is roughly what we expect for 10 year runs.) Note that roughly 95% of test will exhibit slopes above roughly 0.4C/century; only 5% will exhibit slopes less than 0.4C/century.

For the purpose of these test, I’ll chose a confidence level of 95%, which results in a “alpha error” of 5%; i.e. α=0.05.

Note that for the purpose of this test H_o is considered provisionally true. It will not be rejected unless the results we get are inconsistent with (1) and that result would have happened less than 5% of the time due to random variations in weather noise and measurement noise.

That is: I reject if the one experiment I do falls to the left of a ‘critical’ slope, similar to the one drawn in the figure above.

I will use the term “falsified” the the statistical test resulted in the conclusion “reject H_o as inconsistent with the data.”

Note that if the hypothesis in (1) really is correct, we expect we will mistakenly reject it α=5% of the time. (When we ‘falsify’ a hypothesis that is actually true, we call this alpha error. I’ll discuss beta error, some time later.)

Why Ordinary Least Squares?

To test the hypothesis in (1) I will use an ordinary least squares (OLS) to fit a straight line to GISS Land/Ocean annual average data for the N=10 years from 2008-2017, inclusive. For current purposes, I will be neglecting serial autocorrelation in the residuals. (People can argue about that later.)

I’m using OLS because many people are familiar with it; it can be done on Excel or using any number of other packages. To apply this method, I will fit N=10 pairs of (temperature, year) data (i.e. (T_i,Y_i) to an equation of the form:

(2)T = m_s Y + b_s

where the sample slope and intercept (i.e. m_s and b_s) are selected such that the sum of the squares of residuals are minimized using the data sample at hand. That is, when doing this calculation, we minimize the the sum of the squares of residuals, “e_i“, where

(3)e_i=T_i– (m_s Y_i + b_s)

A residual on an OLS for the most recent 10 years of Land/Ocean data is illustrated in blue in the figure above and to the left.

Why GISS Land/Ocean?

I will use GISS Land/Ocean data because it’s an easily available product, constantly under surveillance by both NASA and skeptics groups, and NASA is more open about data than Hadley. I’m using annual averages because the serial residuals in OLS for monthly data contain huge amounts of autocorrelation, which complicates the analysis.

Results from Historical OLS

To find the lower bounds on a sample slope “m_{s“, I will need an estimate of the scatter about the best fit line that I will get when I do the experiment; that is, I need an estimate for σm.}

To estimate this value, I will first need two other bits of information. These are:

1. σ_Y, the standard deviation for the number of years in a sample. Because we will be acquiring data for ‘N’ years, this is calculated using (Σi²)/N – {(Σi)/N}², where Σ represents the sum over ‘i’ from 1 to N. This is just a function of N; For N=2, this is equal to ( 5/2 – (3/2)^2 ) = 0.25;

   I wrote a little Excel macro to calculate the value as a function of N.

2. σ_T. standard residual error between temperature data and any OLS fit to data. In principle, for any stationary process, this does not vary in time.

   If one has data, and does a fit, it is easily obtain an estimate for σ_T; the estimate is called the sample residual error, and is denoted s_T. All one does fine e_i for each data point in the fit, square them, sum them and divide by (N-1); then take the square root. (This value is output by LINEST in excel and listed as “s_T“. This gives the unbiased estimator of σ_T.

   However, when planning an experiment, I can’t calculate s_T. I will need to guess or estimate σ_T based on something else. I will do this based on historic data.

   OLS to historic data can be easily performed. If I assume AGW is wrong, I could fit a straight line to data since 1880; in this case I get σ_T=0.0125 C.

   However, if I assume AGW is correct — as is consistent with assuming the hypothesis in (1) is true, I notice that there is a large kink in the data after CO₂ started rising in the atmosphere. So, rather than fit a line to data since 1880, I can fit a line to data from 1970 to the present. I find the slope is m=1.7 C/century and the residual errors σ_T=0.098 C.

   I rounded this to σ_T~0.1C. Once determined, this is considered a constant for all future calculations.

   (Note: I did not account for serial autocorrelation in the residuals here. Serial autocorrelation would raise my estimate for σ_T based on fits to historical data.)

The standard error in “m”, σ_m for experiment with “N” data pairs can now be estimated using:

(4)σ_m² = (σ_T²/(N σ_Y²)

where N is the number of data pairs in my sample; for 10 years, N=10. (Note, this formula differs slightly compared to calculating s_m based on a data sample. This is because I am using an assumed known value of σ_Y, not a sample value, s_Y, in the equation.)

For the values I have selected: σ_Y=0.1C, and N=10, I get σ_m ~ 0.011 C/year, or 1.1C/century.

I will be using this value for the 10 year estimates.

Calculate cut-off m_crit for the upcoming 10 years.

I can now estimate the minimum m_crit that is consistent with 2C a century; this “m_crit” would correspond to the vertical line in the hypothetical normal distribution in figure X above.

However, I must, at this point, I must find the value of a multiplicative constant corresponding to point 95% confidence interval for the statistical distribution of either of slopes ‘m’ for all possible future experiments. If I were running many, many experiments, I would not need to make any choice.

I would know that, if the predicted slope, m_p, is true, then in 95% of possible experiments for the upcoming 10 years, the measured slope must be exceed a critical value m_crit with

(5)m_crit = m_p– 1.646σ_m

So, if the measured value of m_s is less than m_crit, the claim m_pcrit=1.646 is the value of the ‘t’ coefficient for the 95% confidence interval for a cumulative normal distribution. This is the appropriate value for a one-tailed test as I am doing here.

If I used that value, I would conclude that if the sample slope measured for the years 2008-2019 is less than m_crit=0.2 C/century, the IPCC projection of 2C/century is falsified.

Note however, that in yesterday’s post, I put the cut off at m_crit=0.0C/century.

My lower cut-off of 0.0C/century makes it more difficult to falsify the IPCC projection. (If you prefer the 0.2C/century, you can find plenty of people who would prefer it in non-tendentious, non-political settings.)

However, here’s the difficulty: I want a threshold for m_crit that is firm, and not subject to arguments over this “1.646” value. And unfortunately, a different analyst can advance the following point of view:

If we knew for sure that the true value of σ_T, the standard residual error in “T” for any curve fit was 0.1C and that the only reason σ_T appears to vary from experiment to experiment is weather noise, we could also use the multiplicative factor of 1.646 from a normal distribution function.

However, the reality is, we can’t be certain σ_T is a constant; itmay change as a result of AGW. So, while I can estimate σ_T from historical data, one may argue I should assume that the sample value I obtain in my future curve fit using future data the best estimate for σ_T.

This means that instead of using the normal distribution to obtain the constant t_crit=1.64, I will, instead look up the inverse “T” distribution for “N-2” data points. Because I am doing a one sided sample test, I multiply the confidence α by two, and get TINV(0.10, 10-2) = 1.860, using Excel. (Note, if we waited a thousand years, TINV(0.10, 1,000) = 1.646, as required.)

So, using t= 1.86 for N=10 years, I get the following critical value for the slope.

m_crit = m_p– 1.86 σ_m
m_crit = 2.0 C/century – 1.86 * 1.1 C/century = 0 C/century.

So, if the sample slope falls below 0 C/century based on 10 year data, I consider the IPCC projection of 2.0 C/century or above, falsified.

Of course, this test doesn’t won’t falsify the lower range of the IPCC projections. Details describing values of m_crit for various projected values of warming were provided in yesterday’s post.

Conclusion:

If, the average trend for the years 2008-2017 is negative, then any trend of 2.0 C/year or higher, will be found to be “inconsistent with the measured trend” to the 95% confidence level. That is: current projections of that warming may be 2.0 C/year or higher will be falsified.

The analysis was done making one choice that tends to make falsification moredifficult: I picked a ‘t’ value of 1.86 rather than 1.646. I also neglected serial autocorrelation in residuals: this makes it a little easier to falsify.

Results for analyses using 5, 8 or 10 years, and other temperature trends are provided in my previous post.