Archaeologists Beware!

Here we have the latest technology - a trap to catch archaeologists:



How does it work, you are asking yourselves. Well, it's based on the observation that archaeologists are fascinated by old rubbish. So, the trap is set up to mimic a mediaeval rubbish bin. The archaeologist is immediately attracted, and leaps in. When he is in there up to the middens, the trapper leaps out and ties up the archaeologist.

The one question I cannot answer is why anyone would want to capture archaeologists. Pest control?
Read more!

stRing

Here is some more code, to keep the troll confused and happy.


library(ggm)

# Figure 3.1 data
Data=read.table("http://pages.usherbrooke.ca/jshipley/recherche/NINA%20files/fig31.dat")

# Write the DAG
DAG1=DAG(V2~V1, V3~V2, V4~V2, V5~V3, V5~V4, V5~V6)
# Draw the DAG
drawGraph(DAG1)
# D-separation test
dSep(DAG1, "V3","V1", NULL)
# Shipley Test
shipley.test(DAG1, cov(Data), dim(Data)[1])

#####################################################
#### SEM

Data=read.table("http://pages.usherbrooke.ca/jshipley/recherche/NINA%20files/fig31.dat")
names(Data)=LETTERS[1:dim(Data)[2]]

library(sem)
# Input the model
path3.1=specify.model()
A -> B, beta.AB, NA
B -> C, beta.BC, NA
B -> D, beta.BD, NA
C -> E, beta.CE, NA
D -> E, beta.DE, NA
F -> E, beta.FE, NA
A <-> A, var.A, NA
B <-> B, var.B, NA
C <-> C, var.C, NA
D <-> D, var.D, NA
E <-> E, var.E, NA
F <-> F, var.F, NA

# Fit the SEM
model3.1=sem(path3.1, cov(Data), N=dim(Data)[1])
summary(model3.1)

# Specify an alternative model: remove the B->D path
path3.2=specify.model()
A -> B, beta.AB, NA
B -> C, beta.BC, NA
C -> E, beta.CE, NA
D -> E, beta.DE, NA
F -> E, beta.FE, NA
A <-> A, var.A, NA
B <-> B, var.B, NA
C <-> C, var.C, NA
D <-> D, var.D, NA
E <-> E, var.E, NA
F <-> F, var.F, NA

path3.2

model3.2=sem(path3.2, cov(Data), N=dim(Data)[1])
summary(model3.2)

# Compare the two models
anova(model3.1, model3.2)

###################
# String

# Read data (use your own file!)
string=read.table("string.txt")

# First parameterisation
path.string1=specify.model()
Length -> V1, NA, 1
Length -> V2, beta.2, NA
Length -> V3, beta.3, NA
Length -> V4, beta.4, NA
V1 <-> V1, var.1, NA
V2 <-> V2, var.2, NA
V3 <-> V3, var.3, NA
V4 <-> V4, var.4, NA
Length <-> Length, var.L, NA

model.string1=sem(path.string1, cov(string), N=dim(string)[1])
summary(model.string1)

# First parameterisation, Var(V1) set to zero
path.string1a=specify.model()
Length -> V1, NA, 1
Length -> V2, beta.2, NA
Length -> V3, beta.3, NA
Length -> V4, beta.4, NA
V1 <-> V1, NA, 0
V2 <-> V2, var.2, NA
V3 <-> V3, var.3, NA
V4 <-> V4, var.4, NA
Length <-> Length, var.L, NA

model.string1a=sem(path.string1a, cov(string), N=dim(string)[1])
summary(model.string1a)

# Second parameterisation
path.string2=specify.model()
Length -> V1, beta.1, NA
Length -> V2, beta.2, NA
Length -> V3, beta.3, NA
Length -> V4, beta.4, NA
V1 <-> V1, var.1, NA
V2 <-> V2, var.2, NA
V3 <-> V3, var.3, NA
V4 <-> V4, var.4, NA
Length <-> Length, NA, 1

model.string2=sem(path.string2, cov(string), N=dim(string)[1])
summary(model.string2)

# Second parameterisation, Var(V1) set to zero
path.string2a=specify.model()
Length -> V1, beta.1, NA
Length -> V2, beta.2, NA
Length -> V3, beta.3, NA
Length -> V4, beta.4, NA
V1 <-> V1, NA, 0
V2 <-> V2, var.2, NA
V3 <-> V3, var.3, NA
V4 <-> V4, var.4, NA
Length <-> Length, NA, 1

model.string2a=sem(path.string2a, cov(string), N=dim(string)[1])
summary(model.string2a)

# Profile likelihood, to look at likelihood of Var(V1)
GetDev=function(var, path, Cov, N, ...) {
path[5,3] = var
deviance(sem(path, S=Cov, N=N, ...))
}

Var1=as.character(seq(0,5, length=100))
Deviance=sapply(Var1, GetDev, path=path.string1a, Cov=cov(string), N=dim(string)[1])

plot(Var1, -Deviance+min(Deviance), type="l")
abline(h=-3.71, col=2)

DevDiff=-Deviance+min(Deviance)
max(Var1[which(DevDiff>-3.71)])
Read more!

How long is a piece of string?

Possibly the best post I will ever make....

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34.86415 10.10497 12.8474 31.22481
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53.4557 7.246561 27.43898 43.97807
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51.66953 7.324016 23.63456 42.91175
66.42632 8.658774 29.22023 72.33342
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56.01276 14.57870 22.03388 53.55534
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56.74827 3.794125 20.60873 68.87886
45.77865 12.46980 21.70479 36.5648
46.62047 4.502308 11.93615 54.10634
41.05144 5.667886 12.01179 41.78962
53.2453 -3.868202 16.93013 50.27663
53.7888 11.31978 17.11798 56.03127
30.87982 3.827967 11.30715 39.13347
61.44193 3.093164 21.16498 55.60321
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37.31856 8.498543 13.68324 34.73792
42.86177 8.966008 21.06236 25.42074
46.94618 11.26505 18.59283 49.20128
30.25900 -9.73547 12.67027 17.03394
38.62156 -4.717086 15.84583 41.32916
61.35311 -4.765051 26.90832 51.92822
49.59635 3.047772 23.48041 51.40419
52.02974 2.293412 20.37837 67.17684
50.19866 2.375115 19.91393 57.71842
46.03202 -4.013739 17.67725 61.51833
56.86154 14.15691 22.48793 43.30352
56.41994 -7.764509 19.64890 53.54802
54.06958 -3.601526 25.26686 59.35605
56.83714 -9.573361 23.74937 57.70736
53.64752 2.096009 20.35184 57.93559
37.27191 15.30242 20.21106 33.38579
59.88905 11.07137 17.94779 71.80114
55.42092 -9.012312 18.61974 54.39239
39.05826 -2.005509 14.99925 54.7657
39.92606 6.116922 17.55380 15.37107
35.33501 6.035102 18.93004 33.78016
43.38075 -0.8498992 11.55096 31.35522
51.15989 3.137947 17.99312 60.41017
55.67945 -3.199316 26.5845 68.93158
44.06857 4.256001 16.96168 55.44931
72.77981 21.3873 29.13402 63.98076
62.06794 -2.856957 23.64693 58.15178
65.27273 9.363268 27.89871 79.48394
55.48338 1.583964 28.82452 60.35442
49.26494 3.065723 23.86022 35.78034
37.16467 -3.88511 9.301194 32.40567
54.7703 15.03396 21.67751 37.43552
58.50714 13.30801 19.32692 67.2622
40.02551 10.48335 9.715556 21.38574
46.17735 11.74318 16.51314 45.3918
50.65409 -0.3688263 15.42613 43.45416
36.85831 7.61677 12.41689 40.44595
65.48127 15.54783 30.48558 60.1603
43.28649 -4.36009 16.61700 40.53279
47.33055 -9.820944 22.52366 45.69401
60.24609 15.18745 22.2615 80.29985
47.6109 6.291483 18.42406 46.20005
32.52579 2.163241 9.673215 35.27061
48.10742 5.023338 28.71845 44.01503
42.53343 10.37183 22.23493 51.5103
51.65264 0.7802082 15.98401 53.49945
73.92167 7.71875 29.47882 65.73937
50.08472 7.385094 14.87745 50.5794
31.05070 5.02159 6.320998 47.83895
50.46649 3.390473 17.70616 48.57303
65.88473 15.99298 27.95033 48.00119
71.943 20.16625 26.64881 93.15871
50.15365 -5.470214 21.19497 51.69552
53.1029 4.348359 20.93293 25.96702
57.25345 5.215286 19.01661 43.09172
47.21926 -18.31202 19.56487 44.35467
51.86111 -2.302820 23.37832 34.78882
40.86635 3.55813 11.12562 59.46731
41.81289 1.341042 8.254875 53.02104
66.34576 4.365926 29.19620 53.34384
53.60574 -2.281706 25.03334 53.4722
53.36986 5.65597 15.21477 60.28249
44.3015 10.90607 16.77324 63.7355
42.04445 1.970218 22.04973 44.00357
52.80729 0.551982 15.49625 54.89782
59.94559 9.41296 19.90171 45.41671
55.74521 16.31075 26.3625 60.29774
37.11364 1.525146 14.04184 34.33557
32.83712 2.774706 6.601578 41.34569
43.33734 -3.330938 19.47048 38.48738
47.11865 8.52896 23.66342 62.64756
59.86298 0.1594297 26.43365 72.1182
47.38146 11.47643 17.11502 56.6119
50.51492 -6.982039 21.46275 57.37549
41.04778 2.430464 11.45823 31.58703
47.56391 8.033224 18.18477 52.83157
38.83323 5.135054 18.99735 33.14404
33.55940 4.576157 11.44899 28.96809
65.17481 9.532619 27.40959 53.81124
60.97809 10.72855 23.05310 49.10621
65.71203 4.680273 22.60413 65.91371
44.6671 12.72787 15.06389 37.87554
46.70985 4.207429 15.65107 49.61584
44.0126 -0.8969562 24.53429 37.99542
56.09004 6.60307 19.38070 62.44445
52.74047 2.275871 22.03446 48.78008
43.1794 -1.274556 18.38257 52.86725
52.52871 4.785603 19.78508 55.5101
50.91668 8.13449 20.70597 44.02226
53.04344 22.67535 21.92497 42.16448
54.31046 -5.32972 21.98447 64.63553
51.76945 4.045078 20.51839 52.57333
57.46368 4.722063 26.79969 67.60241
48.26568 0.9973341 19.01231 43.01635
39.52084 3.624995 20.03794 19.2539
56.47525 15.15424 18.85206 54.79408
64.2143 7.907288 25.63994 69.30486
46.54991 -5.522777 21.88838 50.19121
48.57942 9.917584 5.490708 58.09242
68.94336 15.78718 32.37786 71.80714
65.097 5.148921 30.7213 84.25914
55.61453 2.618225 14.70006 58.49522
43.70043 2.583815 21.14595 39.04892
62.93275 17.63986 25.88950 75.09722
40.43976 -0.585903 13.92127 36.75159
55.45437 -0.3992024 24.08535 36.08245
43.35162 9.897227 14.66669 42.22049
47.67736 -0.7712326 15.27691 45.10154
33.92403 8.630114 16.55974 31.888
54.21994 2.466792 20.30346 55.51824
44.68194 8.934485 15.75736 23.64034
51.26517 1.499947 22.02061 39.83077
37.44306 -5.439318 8.345963 38.25729
44.72494 5.445372 22.24458 67.62202
50.8323 8.911907 20.07359 57.16967
55.58463 1.375984 24.00998 58.9103
51.32851 12.48013 14.64975 36.59635
50.01345 3.827587 20.03792 49.61945
39.40343 14.19370 15.54872 45.10012
44.0108 -6.723932 14.32292 61.41351
44.85873 10.95146 16.78062 51.02501
45.62154 13.41332 20.0166 47.36642
46.49261 3.184934 21.42679 63.42772
21.25775 10.71705 10.84546 33.68133
72.93982 8.190522 27.71297 75.4493
65.63378 5.692074 27.75515 55.73338
47.33759 3.022367 13.15211 44.59232
38.77872 6.00241 19.44568 48.34907
63.15298 -3.808 30.82053 80.11868
43.96191 -3.850284 19.4795 43.23053
44.00066 9.85134 20.63579 55.01284
50.88649 14.63406 23.84008 44.95536
50.22891 -4.04665 20.18481 32.5259
43.60246 -1.379428 16.41767 40.18411
48.53157 5.588242 18.69031 61.24137
41.77733 1.084981 19.79319 43.38450
58.40972 8.500872 24.77230 49.15017
57.57225 -3.797773 22.58461 69.06964
40.20747 -4.917563 13.88889 47.72803
51.50455 -9.029045 18.24453 46.40172
51.22361 0.7188372 19.85480 44.53326
42.90079 10.53299 18.18231 35.82266
54.97448 6.566912 21.16093 38.28001
44.30551 18.52872 16.14252 29.35241
42.90053 11.1345 20.60380 47.55508
40.59100 3.836315 19.17098 40.01481
64.46993 3.410776 19.69802 52.40692
46.05315 3.149181 15.95172 37.64019
38.9236 13.07707 16.62506 38.5357
50.07179 -2.132767 10.71973 53.55615
62.8026 3.477998 22.30143 61.35994
29.33549 -2.266676 8.703178 28.27489
52.82056 -5.803662 20.60624 63.00311
39.763 8.74093 11.52920 48.11935
49.20929 10.07257 18.14720 47.40108
59.08706 12.83303 28.72768 63.30883
48.69612 5.591273 22.10258 37.92279
45.22457 7.19037 18.16479 29.70533
58.29379 7.562675 21.38561 56.18841
60.23416 0.4790846 25.61319 54.84405
40.80870 14.72674 16.83701 52.38186
44.07144 5.012861 21.17531 41.13701
41.01263 -6.398877 21.61346 39.50615
57.26699 12.37304 21.68357 66.50561
45.10022 3.784656 17.50116 46.27478
47.70641 -1.740854 21.81915 49.36876
49.49623 7.38965 13.35020 53.29553
67.39344 0.3582493 32.50292 83.45366
43.91011 6.955515 20.1076 42.81388
51.06955 11.46009 27.47350 61.35225
70.06126 11.46235 27.73504 85.74356
62.28284 7.52419 30.89928 64.75573
46.35349 12.13339 13.76977 37.60365
61.63559 4.779976 28.43432 80.95307
49.11642 7.717767 23.40764 62.98167
29.38953 15.28556 8.171793 24.36978
59.50431 0.1827939 24.80716 54.25433
47.56914 -5.181217 17.58222 43.88487
49.92213 7.02148 15.47130 61.99712
49.03364 4.933716 12.89841 39.03318
45.31894 8.319285 18.97337 47.01734
46.90397 10.42634 17.42944 48.40244
38.53745 -3.053576 12.14128 38.62005
53.53033 -13.54362 25.11791 49.2909
43.88261 -3.446498 16.81401 51.68662
56.41206 10.91076 23.93817 59.53211
50.15494 9.85292 18.61940 53.80352
53.84678 -1.913277 20.18228 43.39612
54.07992 8.641245 20.06299 44.45265
47.44463 2.684312 18.93508 48.09456
47.47627 10.80877 17.72176 58.99301
39.82833 8.556109 14.18629 55.5331
33.24262 -4.679348 13.79564 20.6876
53.47825 2.301145 19.61014 38.3461
43.7571 10.22414 15.54398 54.45097
39.99939 -2.742248 22.66244 37.48000
62.0932 2.095559 24.75525 69.32196
25.84546 2.086693 6.911238 34.54912
54.73739 -8.796734 15.48919 43.64274
46.72121 11.00066 15.6844 52.25666
47.94386 11.10268 18.31938 44.31964
43.60601 -6.907512 12.91284 45.1927
60.80403 -4.68798 22.13393 60.51706
42.9251 8.836826 16.09451 55.54047
41.19792 7.915696 20.71839 47.43137
39.85169 -4.470165 16.78824 50.02049
44.74882 6.133915 17.40697 45.12814
51.0531 1.828886 16.24041 64.59942
42.4202 3.107667 12.01330 52.86244
54.34774 -11.8652 23.02954 52.83223
35.47003 12.79844 17.7721 60.13769
49.39028 -13.65518 22.57524 43.38453
43.71669 -6.957278 17.17567 46.32105
58.35255 9.28393 23.99462 57.64893
76.23737 9.186005 30.78686 79.4134
46.33435 16.42037 14.99501 41.66136
51.12072 3.630508 14.11113 66.11836
34.31019 1.043082 14.72027 36.24138
51.97107 7.015492 14.57453 66.39966
62.43678 7.683385 26.39867 55.5293
57.27632 1.404660 22.24053 70.24073
49.5025 7.503132 22.38669 33.98815
45.59278 -1.347406 18.73675 44.78555
28.63468 0.8145263 14.24592 15.69798
35.79533 -12.44964 13.90434 28.42654
28.59595 7.324136 12.97275 32.70563
52.87595 -5.731458 23.34969 33.35862
43.8342 12.19212 15.74296 47.8095
50.24297 4.486603 15.69612 53.69657
50.53234 -11.37676 20.09458 37.95678
40.65528 -1.754301 21.72338 29.18788
50.68405 1.010595 18.3277 56.25837
39.82078 -2.874696 14.28346 28.24900
53.22794 2.402170 23.42319 51.68007

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Statistical Modelling: The Bits Pt. 2

In my last post, I outlined the ideas of statistical inference: that we have a model, and we have a philosophical scheme that tells us what maths to do to estimate the parameters of the model. But this is not the same as doing this in practice. The equations are often complex, and difficult or impossible to solve (anyone fancy trying to integrate in 1000 dimensions?). Because of this, a lot of methods have been devised for estimating the parameters, and particularly estimating their variability (which, from now on, I will write about as estimating the standard error).

For maximum likelihood, the problem of getting the estimates is an optimisation problem: find the set of parameters that give the largest likelihood. For the simple linear regression, this is easy, because the equations were worked out a couple of hundred years ago, by some French mathematicians trying to work out where Paris was. For more complex models, such as generalised linear models, there are algorithms that have been shown to efficiently iterate towards the ML estimates. In more complex models, more general or complicated search algorithms might be needed (for example, there is the EM algorithm which is useful for dealing with missing data. So useful that the missing data is sometimes invented, in order to use it).

These algorithms only give point estimates. We also want to know the standard error, and there are methods for doing this. Again, it may be that the equations can be derived directly, or can be found with a simple numerical search. For less standard problems, more complex algorithms can be used. Two popular ones are called the bootstrap and the jackknife. They both work by re-sampling the data, and using that variation to estimate the standard error. What is less well known is that they can also be used to estimate the bias in the point estimates.

Even more complex problems have even more complex solutions. For Bayesians, the most common method for fitting their models is a technique called MCMC. This estimates the parameters (actually their full distribution) by simulation. The technique is not simple to explain, but it is used because it is very flexible and for a lot of problems works well in practice. However, this is not the same as a Bayesian analysis: MCMC can be used by frequentists as well, and other complicated methods can also be used to fit Bayesian models (e.g. sequential importance sampling).

So, now we have a model fitted. There is still one part of the process that is often done: model selection. This is not always done, and is perhaps done more often than it should. It also creates a lot of heated debates, which may surprise some of you. That's because model selection is often disguised as hypothesis testing.

A general description of the role of model selection is that it is a way of choosing between different mathematical models for the data, to find the best one. For frequentists there are two principal ways of doing this: null hypothesis statistical testing and using information criteria to compare models.

Null hypothesis statistical testing (NHST) is the method everyone is taught to test hypotheses. What one does is set up a null hypothesis, for example that there is no difference between two groups, and then test whether this hypothesis is supported by the data. If it is, then the null hypothesis is accepted (strictly, it's not rejected), otherwise it is rejected. In one common formulation, whenever a null hypothesis is set up, a more general alternative hypothesis is also proposed: rejecting the null hypothesis means accepting the alternative hypothesis.

Where is the model selection in this? We can take the Harry Potter example, and test the null hypothesis that the change in book length over time is zero. The relevant part of the mathematical model is

m_i = a + b X_i

where m_i is the expected number of pages, and X_i is the year of publication. The null hypothesis is that b=0. But this is equivalent to this model

m_i = a

And m_i = a + b X_i is the alternative model for the alternative hypothesis. Hence, hypothesis testing becomes a problem of comparing two models. The models are compared by calculating the amount of variation in the data explained by the two models, and seeing if the larger model (i.e. the one with the effect of time) explains a significantly greater amount of variation.

The alternative method for model comparison uses information criteria. These are measures of model adequacy: they consist of a measure of model fit (the deviance: the smaller the deviance, the closer the fitted model is to the data), and a penalisation term for complexity (the more parameters in the model, the higher this penalisation). The different models being compared are fitted to the data, and the relevant information criterion (e.g. AIC or BIC) is calculated for each one. The model with the lowest criterion is declared the best. If there are several models with similar criteria, they are sometimes all examined, and one of them chosen to be the best, for other reasons (e.g. because it makes sense scientifically).

Both of these methods have their counterparts in Bayesian analysis: hypothesis testing can be carried out using Bayes factors, for instance, and there is an information criterion for Bayesians (DIC. Alas, not the Bayesian information criterion!).

So, there we have it. OF course, this is not all there is in statistical inference: for example, I have not dealt with model checking (i.e. asking whether the model actually fits the data well), and there are many details I have left out. But I hope this give a scheme that separates out the different parts of fitting statistical models, and reduces some of the confusion. To summarise, here is a list of the parts of the process, and examples of the actual methods used in each part:

1. The mathematical model




• regression, ANOVA


• ARIMA (time series)


• Generalised linear models, and Generalised linear mixed models




2. The inferential framework




• Frequentist (Maximum Likelihood and REML)


• Bayesian


• least squares


• non-parametric




3. Model fitting (i.e. the computations)




• bootstrap and jackknife


• MCMC


• importance sampling and sequential importance sampling




4. Model selection




• Significance tests


• information methods (?IC: AIC, BIC, DIC, etc)

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Statistical Modelling: The Bits Pt. 1

It's a sad fact of life that it's too short, so we don't have time to learn everything that we need to know. For most biologists (and probably most scientists, and indeed social scientists), one of the things they should learn to do science well is statistics. Of course, the poor dears are too busy learning about things like biochemistry or PCR, to have time to learn about the important stuff.

Because of this, there is a lot they don't know, and also a lot of confusion about the bits they do know. I therefore thought it was worth writing a few posts about some of these issues, to try and dispel at least some of the confusion.

Now we're below the fold, I should admit that I don't mind too much the lack of knowledge - there are large swathes of statistics that I don't know about either (I was too busy admiring the paper aeroplane building skills of my fellow biochemistry students). The confusion is more of a concern, and I think it is largely because biologists get little training in statistics as undergraduates, so don't have the grounding in the theory when they start doing research. And, yes, I am generalising - some biologists do become very good statisticians. But there is still plenty of confusion.

A lot of basic confusion comes from not understanding the different parts of using models to analyse data. For the moment I am ignoring things like model checking, and how to interpret the results. This is not because they are unimportant, but just because I want to write a blog entry, not a monograph. Anyway, There are, I think, 4 parts that can be separated out:

1. the mathematical model
2. the inferential framework
3. model fitting (i.e. the computations)
4. model selection

Point 1 is often passed over, and points 3 and 4 are often confused with point 2. I will deal with the first two points in this post, and treat the others later.

To help disentangle the different parts, it is useful to have an example. For this, we can use a simple linear regression, such as the one I used here. The mathematical model is self-explanatory. For the regression, it can be written as

m_i = a + b X_i
Y_i ~ N(m_i, s²).

The first line is the equation for a straight line. X_i is the covariate (year, in the example). This is then related to an expected value, m_i. The second line says that each data point, Y_i, is normally distributed with mean m_i and variance s².

Note that the data are assumed to be drawn from some distribution. Hence, they are random. However, they depend on parameters, in this case the mean and variance. Here the mean is modelled further, using an equation, so it is deterministic. i.e. if we knew the parameters, we would know the true value of the mean.

More complex analyses also have a model underneath them: the usual ANOVA actually has a model that is almost the same. The mean can be a function of several parameters, and might not be the mean, but just a parameter that controls the likelihood. The variance might also be modelled. And the parameters that are modelled could themselves be functions of more parameters. And all these parameters might themselves be random. Yes, things can get complex.

The problem, then, is to estimate the parameters. There are several schemes for doing this. For the model above, the two commonly used schemes are the frequentist and the Bayesian approaches. Each of these gives us a way to estimate the parameters. They are, in essence, philosophical approaches to understanding what we mean by probability and randomness. In the frequentist scheme, what we observe is random, with fixed parameters, and hence any statistic we calculate (such as the slope of the straight line) is a random function of the data. In contrast, the Bayesian approach is to say that anything we are uncertain about is random. Hence, we treat the data as fixed (because we know it: we have observed it), and the parameters are random.

Both of these philosophical schemes are useful in that they then can be used to construct methods for estimating the parameters of the mathematical model. Both use the likelihood, i.e. the probability of the data given values of the parameters, albeit in different ways. For example, the frequentist scheme tells us that we should find the values of the parameters which give us the maximum value of the likelihood. Hence, this is often spoken about as a maximum likelihood approach (there is another philosophical scheme which leads to the same equations, but to me it always looks like asking Fisher to wave his fiducial wand to make everything mysteriously right. I assume I'm missing something important).

Estimating the parameters is not enough, though. We also want to know how reliable they are. It makes a difference to our estimate that for every year J.K. Rowling writes an extra 300 pages if the range of possible values is 290-310 or 0-600. In the former case, we have a pretty good idea how much time to book off work to read her latest book: in the latter, we have little idea about whether we need to have a headache or major heart surgery. We can summarise the reliability in a variety of ways: a popular one is the probability that the estimate is less than zero. Alternatives include giving a range of likely values (typically a range for which there is a 95% probability that the estimate is within it), or a statistic such as a standard deviation to summarise the variability in the estimate (in the frequentist scheme, this standard deviation is called the standard error).

In my next post, I will describe the model fitting and model selection, i.e. how these schemes are used in practice to make inferences. I suspect this will include a rant about p-values.

Before I stop to work myself up into a frenzy, one last point is worth mentioning. The approaches discussed above assume that the data are normally distributed. If we discard this assumption, we can use different approaches. One, called least squares, is based on optimising a property of the model and data (i.e. minimising the sum of the squares of the differences between the data and the expected values). It turns out that this is the same as maximising the likelihood, i.e. it is mathematically the same as the frequentist approach. I, therfore, use the frequentist interpretation of the equations, and it gets round a lot of fiddly problems. Another, called non-parametric statistics, does away with any distributional assumption, but also throws away part of the data: rather than use the data themselves, the analysis concentrates on using the ranks of the data (i.e whether each Y_i is the smallest, second smallest etc.). In some cases, I think this can still have a likelihood interpretation, but I must admit that I haven't checked: it's just a hunch.
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Still here?

Wow! The virtual world is still here!

I'm co-teaching a course with Bill Shipley in (sort-of) central Finland. Yesterday we drove up here, and settled in. I tried to get a wireless connection (I had been told there was one) from my room. My laptop would search and find a weak connection, and then when I tried to connect, it would disappear. Repeatedly. Once it decided to mock me by giving me two networks that I couldn't connect to. It would tell me that they were secure networks, which is a hint that I might need a password. But, being rational, I tried to connect anyway.

This may not be relevant at all, but I've just been reading a book about happiness, which explained that we often do things that we expect will make us happier, but we don't get the thrill at the end. This leads to addictive behaviour: we just can't stop ourselves even if we are objectively aware of the futility.

OK, must stop: I have to check my email.
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