Thayer Watkins
Silicon Valley
& Tornado Alley

Assessing the Statistical Significance
of Temperature Trends for North America

The major problem in analyzing climatic trends is establishing whether an unusual development represents a climatic shift or is within the variability of the existing climate. The matter below involves the temperature record of six 10° latitude bands in the Northern Hemisphere for the period 1975 to 1994. For each latitude band a trend rate in °C per decade is estimated and compared with the standard deviation of the trend estimate. The ratio of trend estimate to its standard deviation, called the t-ratio, is used as measure of the statistically significance of the trend.

The Stochastic Structure of Temperature Series

The temperature change for a region is proportional to the net heat inflow and hence if T(t) represents temperature at time t then

T(t+1) = T(t) + U(t)

Thus the temperature is the cumulative sum of all past value of U(s). If the U(s)'s are random variables (disturbances) then the appropriate statistical procedures are different than if the random disturbance terms are independent. In particular the regression estimate of the trend in temperature is an inefficient method because it gives greater weight to the disturbance terms in the middle of the interval of analysis compared to those at the ends. For more on this topic see Temperature Statistics. For variables which are the cumulative sum of random disturbances the proper statistical analysis should be in terms of the period-to-period changes in the variable. The most efficient estimate of the trend is the simplistic one of the change in the temperature over the interval divided by the number of periods in the interval. In the absence of serial correlation the standard deviation of this trend estimate is the standard deviation of the period-to-period changes in temperature divided by the square root of the number of periods in the interval.

The Data

The data are derived from James K. Angell's article "Variation with Height and Latitude of Radiosonde Temperature Trends in North America, 1975-94," published in the Journal of Climate in August of 1999 (Volume 12, Issue 8). Dr. Angell is highly acclaimed for the accuracy and objectivity of his work at the Air Resouces Laboratory of the National Oceanic and Atmospheric Administration at Silver Springs, Maryland. Dr. Angell's focus was on the radiosonde measurements of the atmospheric temperature above the Earth's surface but he also gave results for surface temperature. Dr. Angell used regression estimates of the temperature trends for six latitude bands. In four out of the six bands the trend was not significantly different from zero. The results in this work are a verification of Dr. Angell's results but taking in account that that temperature is the cumulative sum of random disturbances.

Dr. Angell does not provide the raw data in his article but the numerical values can be obtained to a satisfactory degree of approximation by scaling the values from his graphs. There are likely small differences between the data used below and Dr. Angell's original data but these marginal differences do not affect the results. An equatorial band is not included because Dr. Angell limited his analysis to North America. The first band is from 20°N to 30°N. The results of the analysis are shown below:

The upper graph is for the period to period changes in temperature. These values are computed from the temperature data shown in the lower graph but conceptually this arrangement is to make the point that the basic phenomena is the fluctuating net heat flows that give rise to the temperature changes and the accumulation of these generate the temperatures. The mean value of the changes could be computed by summing up the changes but this sum is more readily available as the difference between the last temperature in the interval and the first. The standard deviation is computed using the individual differences. The mean and standard deviation are given in units of °C per year but they could be multiplied by ten to give them in °C per decade. The trend in the temperature is simply the mean period to period change but this is conventionally expressed as °C per decade. The standard deviation is likewise expressed in °C per decade. The t-ratio shows that the trend is not significantly different from zero at about the 95 percent level of confidence.

The results for the other latitude bands are similar. Some have negative trends and some positve but in each case the t-ratios are less than 2 in magnitude and thus they are not significantly different from zero at the conventional 95 percent level of confidence.

Thus the Arctic band of 70°N to 80°N does not have a trend significantly different from zero. The trend estimate is higher, quite higher than the others but so is the variability of the temperature. The net result is that there is no more reason statistically to consider the temperature trend in the Arctic more significant than those in the other bands.

The graphical display of the above data by latitude bands are shown below. The first graph shows the trend estimates and the standard deviation of those trend estimates. Both variables are in units of C°/decade.

The t-ratio is a dimensionless number. The critical level for the t-ratio to be statistically significantly from zero is about 2. As shown below the t-ratio never gets anywhere near the level of 2.

There are climatological reasons for the temperature to be more variable at the higher latitude bands. However there is also a statistical reason for the greater variable. There is a much smaller global area and land area at the higher latitudes than at the lower latitude levels. If φ is the latitude angle in radians and R is the radius of the Earth then the area enclosed within a latitude band of Δφ centered on a latitude of φ is R²Δφcos(φ). This means that if the measured standard deviation are multiplied by the square root of cosine(φ) the results should be approximately the same. Below is a graph of the results of the computation.

(To be continued.)

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