It has not stopped

this-Cowtan & Way this-Hadcrut4 this-JMA this-NASA-GISS this-NOAAAnd finally the series until end 2013 from Berkeley:

 

this-bk2

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GISS Global temperatures: reporting history

GISS Global Sea and Land Surface Temperatures as they were reported in 1981, 1995, 2003 and today. The line are 5 years averages, or, more accurate, 60 months averages.

GISS-4-times

Sources:

Baselines: for 1981 the baseline is the period of 1880-1980. For others it is 1951-1980. Can be found in the documents.

Conclusion: this diagram shows that Steven Goddard erred here.

The “hiatus”

Giss-2

In this picture we have plotted the 60-months moving average against a background of the monthly averages.

If we want to view this moving average as an indication of the current anomaly, then we have to move it to the right, otherwise the moving averages of the “current anomaly” would contain future values.

Normally the hiatus is illustrated by drawing a straight line from around 1997, and – as this line is almost horizontal -it is used as proof that the warming has stalled. However, it is not so simple. In 1997-1998 the global temperatures suddenly rose to new levels, and from that new level indeed the temperatures stayed more or less the same(*).

Look at the dots in the upper right corner. These higher values are all in a sudden  common after 1998, whereas before 1997 there are none of them except for a couple of outliers. After 1998 more than 50% of all months were warmer than any month before (except those 2 outliers).

What we witness is a steep increase of 0.18 degree C in 5 years time followed by a stationary period. You cannot call that a stagnation IMHO. The next blog item gives a more detailed analysis.

(*) Using only values after 1997 in an autocorrelated series is a statistical blunder.

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Not all numbers are mean

In [1] D. Stapel mentioned a test with 32 students with two attributes: secure / insecure and chose meat / chose vegetarian dish. He gave the following table:

60% || 40%
-----------
20% || 80%

The number theorist in me quickly noticed something funny: these percentages are not possible. For integers n1 + n2 = 32, 60% * n1 and 20% * n2 are never close enough to integers at the same time. I pointed this out to the authors, but got no reply. A week later Stapel’s fraud was published.

This is a simple example of a much more general phenomena for discrete distributions.  Not all values are possible for mean, st.dev., etc. I will give some examples using R-code.

Take N samples with replacement from a probability distribution on the numbers 1 to k. The sum of these samples ranges from N to k*N, so there are 1+(k-1)*N possibilities. On the other hand, the sample mean is a number between 1 and k, and if we write the mean with 2 decimals, there are 1+(k-1)*100 different numbers. If N is small, only N% of all these 2-decimals numbers are possible. In other words, for an integer n <= k are in the interval [n, n+1) precisely N numbers possible. Example test:

m=4.13; N=16;
> round(round(m*N, 0)/N, 2)
[1] 4.12

Let’s try the situation of 16 students that score some items on a scale of 1 to 7 (a Likert scale, very popular in circles of social psychology).

v <- function(N, k, ...){sample(1:k, N, replace = TRUE, ...)}
u <- unique(sapply(1:10000, function(i){mean(v(16,7))}))
round(u[order(u)],2)

This will give a row of  numbers starting with 2.31 2.38 2.44 2.50 2.56 2.62 2.69 2.75 2.81 2.88 2.94 3.00 or similar. These are the plausible possibilities. Using the test above we can make a list of the small N’s that can result in the give means:

> b[100:700]=sapply(100:700, function(i){m=round(i/100, 2); return((1:100)[m == round(round(m*(1:100), 0)/(1:100), 2)]) })
> fN <- function(m){b[100*m]}
> fN(4.13)
[[1]]
[1] 15 23 30 31 38 39 45 46 47 52 53 54 55 60 61
[16] 62 63 67 68 69 70 71 75 76 77 78 79 82 83 84
[31] 85 86 87 89 90 91 92 93 94 95 97 98 99 100

— o —

[1] R. Vonk 2011 circulated an unpublished note about “selfish” meat eaters by D. Stapel (e.a.) to the press (in Dutch) that contained faked numbers.

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