Difference between revisions of "Bayes (Hierarcical Bayesian model)"

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This page describes utilization of Hierarcical Bayesin method. The following page briefly introduces theory behind the method, and illustrates the use the method by examples.

The Bayesian method

The Bayes method is based on the bayesian rule. The following equations can be found from the reference 1, pages 7-8. The Bayesian rule is:

p(θ,y) = p(θ)p(y|θ)

,where p(θ) is prior distribution and on p(y|θ) is distribution acquired from the data. By contidionalizing θ with y (which is known), we get:

p(θ|y) = p(θ)p(y|θ) / p(y)

By conditionalizing this way, we obtain p(?|y), which is prior probability conditionalised by the data. This is also called posterior probability. With these equations, we can adjust presuppositions about the probability density function of the data using data itself. As an end result we obtain upgraded probability density function, which is called posterior density.

Posterior distribution ~ prior distribution x distribution estimated by the data

Hierarcical Bayesian method

We create an example of the method in this article. We study dioxin concentrations in different sea area which we assume to be share a similar form of distribution. In other words, we assume that the concentraions in different areas are samples of the total concentration distribution. Thus, we can use observations of concentrations in all areas to estimate concentrations in one particular area of interest. This means conditionalizing observated values to certain parameters. We assume probabilites to these parameters with new parameters, which are called hyperparameters.

There is an extensive explanation of the idea and the method in reference 1, pages 117-121, and you can find a picture of the method in page 119. In our example, we try to create a model which gives us a relatively accurate concentration estimate for all different areas.

Material and methods

Methods

The method used herein is called hierarcical Bayes method. Theory can be read from reference 1. We used WinBUGS and R as a modelling tool.

http://www.mrc-bsu.cam.ac.uk/bugs/

You can get the program and updates from the address above. They also include quite extensive manuals. You need to copy the text to WinBUGS to a new file, and select decode from tools menu and follow the manual.

Table 1. Names of the variables and parameters in the model.

Variable	Distribution	Nature	Parameters	Definition	Description
alpha.tau	gamma	stocastic	10^-3		Parameter of accuracy
alpha.c	norm	stocastic	(0,10^-6)		Mean
alpha0		calculated		(a.c)-(b.c)*xbar	Starting level
alpha[i]	norm	stocastic	(a.c,a.t)		Constant term
beta.tau	gamma	stocastic	10^-3		Parameter of accuraty for slope
beta.c	norm	stocastic	(0,10^-6)		Mean value of slope
beta[i]	norm	stocastic	(b.c,b.t)		Slope
mu[i,j]		calculated		a[i]+b[i]*(x[j]-xbar)	Estimation for concentration
x[j]		constant		input	Age categories
tau.c	gamma	stocastic	10^-3		Parameter of accuracy
sigma		calculated		1/sqrt(tau.c)	Dispersion term
y[i,j]	norm	stocastic	(mu[i,j],tau.c)		Estimate for concentration without observation

Picture 1. Picture of the model in Win BUGS

Error creating thumbnail: Unable to save thumbnail to destination

Material

The data we used in our example consists of dioxin concentration of herring with ages, sizes and location

We create a model similar to example model in WinBUGS. Variables are analogous as demonstrated below: Variables:

Weight of rat = Concentration of herring Age of rat = Age of herring Number of rat (specimen #) = Location

We ran the model with Win BUGS with herring data, and examined the distribution of variables of interest.In rat example the model estimates distributions for weight for each rat. In our model we estimate concentration distributions for each location.

Results

Descriptive statistics

Kuva 2. Age against concentration in herring

Image:Ikä_vs_poitoisuus_silakka.jpg

Kuva 3.Weight against concentration in herring

Error creating thumbnail: Unable to save thumbnail to destination

Results of a linear regression model, where concentraition is explained with age and weight

Call: lm(formula = pitoisuus ~ ikä + paino, data = data)

Residuals: Min 1Q Median 3Q Max -7.9395 -1.3808 -0.1211 1.2573 6.4274

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.68568 0.78077 -2.159 0.0363 * ikä 1.66347 0.14978 11.106 2.39e-14 *** paino -0.03660 0.01205 -3.038 0.0040 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.575 on 44 degrees of freedom Multiple R-squared: 0.7495, Adjusted R-squared: 0.7381 F-statistic: 65.82 on 2 and 44 DF, p-value: 5.952e-14

Huomataan siis että ikä ja paino merkitseviä selittäjiä.

We notice that both age and weight are significant explanators.

Results of WinBUGS calculations

Tables 2 and 3 represent results of our model. In Y[1,2], the first parameter denotes to concentraion of on year old herring in location one (=1), and the second parameter denotes to age category (=2). Thus, the first parameter stands for location and the second parameter for age. Some values (confidence intervall 2.5%) are less than zero (which is not plausible as concentrations can not be negative), but we can control this by not allowing values below zero in our model. In table 3 there are ages divided into four categories (including three years each) and estimates for concentrations. Median values do not have values below zero but 2.5% CI has negative values. Estimates in table 2 are for missing values only. Medians seem to be quite good estimates but the confidence intervalls span very wide due to lack of data.

Table 2. List of results for herringdata with one year age intervals (WinBUGS).

Error creating thumbnail: Unable to save thumbnail to destination

Table 3. List of results for dioxin concentration in herring with three year age intervall (Win BUGS)

Error creating thumbnail: Unable to save thumbnail to destination

See table 4 for Original data of herring

Syntaxes and iterations

Attached are the code of the model (model1, noodle_1 and data_1). There are age groups 1-4 and data_1 for each age categories, as well as inits (if we don't want to simulate them).

All codes open with WinBUGS program and are updated freely. Simulations are using model1 and data_1 and they are working with inits. Similar model can be found in examples (help menu) vol1 Rars. Setting a limit is done by adding syntax I(0,) to the end of distribution definition.

Running the model:

You need to open three windows. Under model find model specification, under inference find samples, and under model find update. With these, you can run the same calculations as we just did.

You can find the files from:

N:\Huippuyksikko\Tutkimus\R55_PMvsDioxin\MeHgvsOmega-3\mallit\Bayes\Aiemmat ajot ja pähkäilyt

Conculsions

We can compare the results with data and draw conclusions. The median estimates seem to have pretty ok values compared the the original data.

Rat example

This model can be found from WinBUGS help page. Our example is based directly to this model.

Rats: a normal hierarchical model

This example is taken from section 6 of Gelfand et al (1990), and concerns 30 young rats whose weights were measured weekly for five weeks. Part of the data is shown below, where Yij is the weight of the ith rat measured at age xj.

Weights Y_ij of rat i on day x_j

	x_j=8	15	22	29	36
Rat 1	151	199	246	283	320
Rat 2	145	199	149	293	354
Rat 30	153	200	244	288	324

A plot of the 30 growth curves suggests some evidence of downward curvature.

The model is essentially a random effects linear growth curve

Y_ij ~ Normal(a_i + b_i(x_j - x_bar), T_c) a_i ~ Normal(a_c, T_a) b_i ~ Normal(b_c, T_b) where x_bar = 22, and T represents the precision (1/variance) of a normal distribution. We note the absence of a parameter representing correlation between a_i and b_i unlike in Gelfand et al 1990. However, see the Birats example in Volume 2 which does explicitly model the covariance between a_i and b_i. For now, we standardise the x_j's around their mean to reduce dependence between a_i and b_i in their likelihood: in fact for the full balanced data, complete independence is achieved. (Note that, in general, prior independence does not force the posterior distributions to be independent).

a_c , t_a?, b_c , T_b?, T_c are given independent "noninformativepriors, with two alternatives considered for T_a and T_b: prior 1 is uniform on the scale of the standard deviations s_a = 1/sqrt(T_a) and a_b = 1/sqrt(T_b), and prior 2 is a gamma(0.001, 0.001) on the precisions T_a and T_b. Interest particularly focuses on the intercept at zero time (birth), denoted a₀ = a_c - v_cx_bar.

Graphical model for rats example (using prior 1):

BUGS language for rats example:

Note the use of a very flat but conjugate prior for the population effects: a locally uniform prior could also have been used.

Data list(x = c(8.0, 15.0, 22.0, 29.0, 36.0), xbar = 22, N = 30, T = 5, Y = structure( .Data = c(151, 199, 246, 283, 320, 145, 199, 249, 293, 354, 147, 214, 263, 312, 328, 155, 200, 237, 272, 297, 135, 188, 230, 280, 323, 159, 210, 252, 298, 331, 141, 189, 231, 275, 305, 159, 201, 248, 297, 338, 177, 236, 285, 350, 376, 134, 182, 220, 260, 296, 160, 208, 261, 313, 352, 143, 188, 220, 273, 314, 154, 200, 244, 289, 325, 171, 221, 270, 326, 358, 163, 216, 242, 281, 312, 160, 207, 248, 288, 324, 142, 187, 234, 280, 316, 156, 203, 243, 283, 317, 157, 212, 259, 307, 336, 152, 203, 246, 286, 321, 154, 205, 253, 298, 334, 139, 190, 225, 267, 302, 146, 191, 229, 272, 302, 157, 211, 250, 285, 323, 132, 185, 237, 286, 331, 160, 207, 257, 303, 345, 169, 216, 261, 295, 333, 157, 205, 248, 289, 316, 137, 180, 219, 258, 291, 153, 200, 244, 286, 324), .Dim = c(30,5)))

(Note: the response data (Y) for the rats example can also be found in the file ratsy.odc in rectangular format. The covariate data (X) can be found in S-Plus format in file ratsx.odc. To load data from each of these files, focus the window containing the open data file before clicking on "Data" from the "Model" menu.)

Inits1 list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250), beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6),

alpha.c = 150, beta.c = 10, tau.c = 1, sigma.alpha = 1, sigma.beta = 1)

Inits2 list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250), beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6), alpha.c = 150, beta.c = 10, tau.c = 1, tau.alpha = 1, tau.beta = 1)

Results

A 1000 update burn in followed by a further 10000 updates gave the parameter estimates:

node mean sd MC error 2.5% median 97.5% start sample alpha0 106.6 3.65 0.04151 99.43 106.5 113.9 1001 10000 beta.c 6.185 0.1102 0.001294 5.967 6.185 6.404 1001 10000 sigma 6.074 0.4673 0.007724 5.247 6.044 7.068 1001 10000

Using Prior 2 (not recommended) has negligible impact

These results may be compared with Figure 5 of Gelfand et al 1990 --- we note that the mean gradient of independent fitted straight lines is 6.19.

Gelfand et al 1990 also consider the problem of missing data, and delete the last observation of cases 6-10, the last two from 11-20, the last 3 from 21-25 and the last 4 from 26-30. The appropriate data file is obtained by simply replacing data values by NA (see below). The model specification is unchanged, since the distinction between observed and unobserved quantities is made in the data file and not the model specification.

click on one of the arrows to open the data for the missing value analysis

Gelfand et al 1990 focus on the parameter estimates and the predictions for the final 4 observations on rat 26. These predictions are obtained automatically in BUGS by monitoring the relevant Y[ ] nodes. The following estimates were obtained:

We note that our estimate 6.58 of bc is substantially greater than that shown in Figure 6 of Gelfand et al 1990. However, plotting the growth curves indicates some curvature with steeper gradients at the beginning: the mean of the estimated gradients of the reduced data is 6.66, compared to 6.19 for the full data. Hence we are inclined to believe our analysis. The observed weights for rat 26 were 207, 257, 303 and 345, compared to our predictions of 204, 250, 295 and 341.

Table 4. Original data for herring

Age	Weight	Concentration
2	15	1.96
4	23	2.49
6	31	6.58
7	43	13.4
11	71.7	17.2
2	17,6	0.876
3	25.2	2.22
3	34.1	1.99
5	54	1.5
6	137	2.18
2	18	2
2	24	2.52
6	33	8.23
9	52	15.9
11	64	16.6
2	16.3	0.704
4	23.6	2.84
7	34	6.79
9	45	14.5
12	65	13.6
2	17	2.29
3	24	3.67
7	33	10.9
8	42.9	11.2
9	55	17.7
1	15.3	0.733
3	23.5	2.35
5	30	4.59
7	41	6.59
3	15.8	2.02
6	22	4.6
7	29	7.4
7	47	5.03
6	144	7.52
1	19	1.13
3	25	1.186
5	36	5.7
2	13	1.42
5	22	3.25
6	32	4.44
8	34	6.58
10	106	3.13
1	17	0.768
3	24	1.79
2	24	0.782
3	48	1.29
7	196	2.31

Tutkittava asia Panulle:

Summa PBDE pitoisuus

Luokitus ensisijaisesti rasvaprosentin mukaan (vaihtoehtoisesti ikä, paino tai pituus). Jaetaan neljään lohkoon (esim. rasvaprosentti 0-2, 2-4, 4-6, >6)

Kokeile eri kategoriointa (rasvaprosentti, ikä, paino, pituus) mikä parhaiten selittää PBDE -pitoisuuden muuttumista.

Samaan tapaan kuin aiempi silakka-data. Priori-jakauma tulee kaikista PBDE-mittauksista ja rasvapidoisuuden täydentävä data tulee kustakin saman rasvapitoisuusluokan omaavista kaloista.

Data kirjasta Hallikainen et al. 2004. Sivut 17- 36. Kotimaisten järvi- ja merikalan dioksiinien, furaanien, dioksiinien kaltaisten PCB –yhdisteiden ja polybromattujen difenyylieettereiden pitoiuudet. EU-kalat. Elintarvikeviraston julkaisuja. Helsinki 2004.

References

1) Gelman Andrew, Carlin John B., Stern Hal S., and Rubin Donald B. Bayesian data analysis – second edition. Chapman & Hall/CRC 2004.

Difference between revisions of "Bayes (Hierarcical Bayesian model)"

Latest revision as of 19:45, 21 July 2013

Contents

The Bayesian method

Hierarcical Bayesian method

Material and methods

Methods

Material

Results

Descriptive statistics

Results of WinBUGS calculations

Syntaxes and iterations

Conculsions

Rat example

Tutkittava asia Panulle:

See also

References

Navigation menu

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@@ Line 1: / Line 1: @@
-. JOHDANTO
+{{method}}
-.1 BAYESMENETELMÄ
+This page describes utilization of Hierarcical Bayesin method. The following page briefly introduces theory behind the method, and illustrates the use the method by examples.
-Bayes menetelmä perustuu bayesin sääntöön. Seuraavat kaavat löytyvät kirjasta Bayesian Data Analysis second edition sivuilta 7-8.
+== The Bayesian method  ==
-Bayesin sääntö on muotoa.
+The Bayes method is based on the bayesian rule. The following equations can be found from the reference 1, pages 7-8. The Bayesian rule is:
 p(θ,y) = p(θ)p(y|θ)
-Tässä p(θ) on priori jakauma ja p(y|θ) on datasta saatu jakauma. Ehdollistamalla θ:taa y:llä, joka on tunnettu, päästään muotoon.
+,where p(θ) is prior distribution and on p(y|θ) is distribution acquired from the data. By contidionalizing θ with y (which is known), we get:
 p(θ|y) = p(θ)p(y|θ) / p(y)
-Näin ehdollistamalla saadaan p(θ|y), joka on siis selkokielellä prioritodennäköisyys ehdollistettuna dataan. Tätä kutsutaan posterioritiheydeksi. Näillä kaavoilla saadaan siis tarkennettua ennakko-oletuksia datan tiheysfunktiosta (priorijakauma) käyttäen dataa itseään hyväksi. Lopputuloksena on siis päivitetty tiheysfunktio jota kutsutaan posterioritiheydeksi.
+By conditionalizing this way, we obtain p(?|y), which is prior probability conditionalised by the data. This is also called posterior probability. With these equations, we can adjust presuppositions about the probability density function of the data using data itself. As an end result we obtain upgraded probability density function, which is called posterior density.
-Muistisääntö:<br>Posteriorijakauma ~ priorijakauma x datasta estimoitu jakauma
+Posterior distribution ~ prior distribution x distribution estimated by the data
-<br>1.2 HIERARKINEN BAYES
+===Hierarcical Bayesian method  ===
-Hierarkisen bayesin idea on helpoin ymmärtää konkreettisen esimerkin kautta. Meidän tapauksessa havaitaan pitoisuuksia eri alueittain, joiden voidaan olettaa olevan alueittain samankaltaisia. Oletetaan, että pitoisuudet alueittain ovat otoksia alueiden yhteisestä pitoisuus jakaumasta. Näin ollen voimme käyttää havaittuja pitoisuuksia yli alueiden estimoitaessa tietyn alueen pitoisuuksia. Tämä tapahtuu ehdollistamalla havaittavissa olevia arvoja tiettyihin parametreihin. Näille parametreille oletetaan todennäköisyydet uusien parametrien avulla, joita kutsutaan hyperparametreiksi.
+We create an example of the method in this article. We study dioxin concentrations in different sea area which we assume to be share a similar form of distribution. In other words, we assume that the concentraions in different areas are samples of the total concentration distribution. Thus, we can use observations of concentrations in all areas to estimate concentrations in one particular area of interest. This means conditionalizing observated values to certain parameters. We assume probabilites to these parameters with new parameters, which are called hyperparameters.
-Kirjassa Bayesian Data Analysis second edition sivuilla 117-121 löytyy kattava selvennys ideasta ja sivulla 119 on kuva hierarkisesta mallista.
+There is an extensive explanation of the idea and the method in reference 1, pages 117-121, and you can find a picture of the method in page 119. In our example, we try to create a model which gives us a relatively accurate concentration estimate for all different areas.
-Käyttötarkoitus meillä on, että pyritään löytämään keinoja mallintaa tilastollisin keinoin pitoisuuksia kaloista. Tavoitteena saada aikaan malli, jonka avulla pienestäkin datasta saataisiin suhteellisen pitävä arvio koko alueen kalapopulaatiolle.
+== Material and methods  ==
-<br>
+=== Methods  ===
-<br>
+The method used herein is called hierarcical Bayes method. Theory can be read from reference 1. We used WinBUGS and R as a modelling tool.
-. MENETELMÄT JA AINEISTOT
-.1 MENETELMÄT
-Käytettävä menetelmä on hierarkkinen bayes. Teoria kirjasta Bayesian data analysis second edition. Työkaluna WinBUGS ja R.
 http://www.mrc-bsu.cam.ac.uk/bugs/
-Tästä osoitteesta voi hakea päivityksen ja lisenssin. Päivitysohjeet ja lisenssiohjeet kattavat, eli tässä ei pitäisi tulla ongelmia. Tekstin vain kopioi WinBUGS uuteen tiedostoon ja tools valikosta decode kuten ohjeissa mainitaan .
+You can get the program and updates from the address above. They also include quite extensive manuals. You need to copy the text to WinBUGS to a new file, and select decode from tools menu and follow the manual.&nbsp;
-Minkä nimisiä ovat muuttujat mallissa, taulukko (muuttujan nimi ja parametrit)<br>Mallin graafinen kuva (vaikka sama kuva kuin WinBUGS:ssa) muuttujien vuorovaikutussuhteista<br><br>
+Table 1. Names of the variables and parameters in the model.&nbsp;
-Taulukko1. Mallin muuttujien nimet ja parametrit
+{| {{prettytable}}
+|-
+| Variable
+| Distribution
+| Nature
+| Parameters
+| Definition
+| Description
+|-
+| alpha.tau
+| gamma
+| stocastic
+| 10^-3
+|
+| Parameter of accuracy
+|-
+| alpha.c
+| norm
+| stocastic
+| (0,10^-6)
+|
+| Mean
+|-
+| alpha0
+|
+| calculated
+|
+| (a.c)-(b.c)*xbar
+| Starting level
+|-
+| alpha[i]
+| norm
+| stocastic
+| (a.c,a.t)
+|
+| Constant term
+|-
+| beta.tau
+| gamma
+| stocastic
+| 10^-3
+|
+| Parameter of accuraty for slope
+|-
+| beta.c
+| norm
+| stocastic
+| (0,10^-6)
+|
+| Mean value of slope
+|-
+| beta[i]
+| norm
+| stocastic
+| (b.c,b.t)
+|
+| Slope
+|-
+| mu[i,j]
+|
+| calculated
+|
+| a[i]+b[i]*(x[j]-xbar)
+| Estimation for concentration
+|-
+| x[j]
+|
+| constant
+|
+| input
+| Age categories
+|-
+| tau.c
+| gamma
+| stocastic
+| 10^-3
+|
+| Parameter of accuracy
+|-
+| sigma
+|
+| calculated
+|
+| 1/sqrt(tau.c)
+| Dispersion term
+|-
+| y[i,j]
+| norm
+| stocastic
+| (mu[i,j],tau.c)
+|
+| Estimate for concentration without&nbsp; observation&nbsp;
+|}
-<br>
-Kuva 1. Mallin graafinen kuva WinBUGS:ssa<br>
+Picture 1. Picture of the model in Win BUGS
-<br>
+[[Image:Bayes diagram.jpg|Image:Bayes_diagram.jpg]]
-.2. AINEISTO
-Datassa on silakan kokoja, ikiä sekä arvoja dioksiini-pitoisuuksille (pg/g WHOPCDD/F)
-<br>Ajatellaan pitoisuutta kalassa, hieman samaan tyyliin, kuin rotta esimerkissä on ajateltu painoa, eli malli on hyvin samankaltainen.
-Muuttujat:
+== Material  ==
-Rotan paino = Silakan pitoisuus<br>Rotan ikä = Silakan ikä<br>Monesko rotta = Mistä pyydetty
+The data we used in our example consists of dioxin concentration of herring with ages, sizes and location
-Tämän jälkeen WinBUGS:lla voi pyörittää rotta mallia silakka-datalla. Näin saadaan mielenkiintoiset muuttujat ja jakaumat esille.
-Rotta esimerkissä siis estimoidaan jakaumat painolla jokaiselle rotalle erikseen. Meillä estimoidaan jakaumat jokaiselle pitoisuudelle alueittain. <br><br>
+We create a model similar to example model in WinBUGS. Variables are analogous as demonstrated below: Variables:
-. TULOKSET
-.1 KUVAILEVIA TILASTOLLISIA TULOKSIA
+Weight of rat = Concentration of herring Age of rat = Age of herring Number of rat (specimen #) = Location
-<br>Kuva 2. Silakan ikä pitoisuuden suhteen.<br><br>
-Kuva 3. Silakan paino pituuden suhteen.<br>
+We ran the model with Win BUGS with herring data, and examined the distribution of variables of interest.In rat example the model estimates distributions for weight for each rat. In our model we estimate concentration distributions for each location.
-<br>
-Lineaarisen regressiomallin tuloksia, joissa pitoisuutta selitetään iällä ja painolla.
+== Results  ==
-Call:<br>lm(formula = pitoisuus ~ ikä + paino, data = data)
+=== Descriptive statistics  ===
-Residuals:<br> Min 1Q Median 3Q Max <br>-7.9395 -1.3808 -0.1211 1.2573 6.4274
+ Kuva 2. Age against concentration in herring
-Coefficients:<br> Estimate Std. Error t value Pr(&gt;|t|) <br>(Intercept) -1.68568 0.78077 -2.159 0.0363 * <br>ikä 1.66347 0.14978 11.106 2.39e-14 ***<br>paino -0.03660 0.01205 -3.038 0.0040 ** <br>---<br>Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+[[Image:Ikä vs poitoisuus silakka.jpg|Image:Ikä_vs_poitoisuus_silakka.jpg]]
-Residual standard error: 2.575 on 44 degrees of freedom<br>Multiple R-squared: 0.7495, Adjusted R-squared: 0.7381 <br>F-statistic: 65.82 on 2 and 44 DF, p-value: 5.952e-14
-Huomataan siis että ikä ja paino merkitseviä selittäjiä.
+Kuva 3.Weight against concentration in herring
+[[Image:Paino vs pitoisuus silakka.jpg|Image:Paino_vs_pitoisuus_silakka.jpg]]
+Results of a linear regression model, where concentraition is explained with age and weight
+Call: lm(formula = pitoisuus ~ ikä + paino, data = data)
+Residuals:  Min 1Q Median 3Q Max  -7.9395 -1.3808 -0.1211 1.2573 6.4274
+Coefficients:  Estimate Std. Error t value Pr(&gt;|t|)  (Intercept) -1.68568 0.78077 -2.159 0.0363 *  ikä 1.66347 0.14978 11.106 2.39e-14 *** paino -0.03660 0.01205 -3.038 0.0040 **  --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+Residual standard error: 2.575 on 44 degrees of freedom Multiple R-squared: 0.7495, Adjusted R-squared: 0.7381  F-statistic: 65.82 on 2 and 44 DF, p-value: 5.952e-14
+Huomataan siis että ikä ja paino merkitseviä selittäjiä.
+We notice that both age and weight are significant explanators.
+===  Results of WinBUGS calculations  ===
+Tables 2 and 3 represent results of our model. In Y[1,2], the first parameter denotes to concentraion of on year old herring in location one (=1), and the second parameter denotes to age category (=2). Thus, the first parameter stands for location and the second parameter for age. Some values (confidence intervall 2.5%) are less than zero (which is not plausible as concentrations can not be negative), but we can control this by not allowing values below zero in our model. In table 3 there are ages divided into four categories (including three years each) and estimates for concentrations.&nbsp; Median values do not have values below zero but 2.5% CI has negative values. Estimates in table 2 are for missing values only. Medians seem to be quite good estimates but the confidence intervalls span very wide due to lack of data.
+Table 2. List of results for herringdata with one year age intervals (WinBUGS).&nbsp;  [[Image:Results1.jpg]]
+Table 3. List of results for dioxin concentration in herring with three year age intervall (Win BUGS)
+[[Image:Results2.jpg]]
+See table 4 for Original data of herring
+=== Syntaxes and iterations&nbsp;  ===
+Attached are the code of the model (model1, noodle_1 and data_1). There are age groups 1-4 and data_1 for each age categories, as well as inits (if we don't want to simulate them).
+All codes open with WinBUGS program and are updated freely. Simulations are using model1 and data_1 and they are working with inits. Similar model can be found in examples (help menu)&nbsp; vol1 Rars. Setting a limit is done by adding syntax I(0,) to the end of distribution definition.
+Running the model:
+You need to open three windows. Under model find model specification, under inference find samples, and under model find update. With these, you can run the same calculations as we just did.
+You can find the files from:
+N:\Huippuyksikko\Tutkimus\R55_PMvsDioxin\MeHgvsOmega-3\mallit\Bayes\Aiemmat ajot ja pähkäilyt
-<br>3.2 WinBUGS ohjelmistolla laskettuja tuloksia.
+===  Conculsions  ===
-Seuraavissa taulukoissa 2 ja 3 esitetään hierarkinen bayes malli silakka-datalle. Huomaamisen arvoista on, että Y[1,1] on ensimmäisen alueen 1 vuotiaan silakan pitoisuus jne. Ensimmäinen indeksi on siis paikka indeksi ja toinen on ikä indeksi. Osa arvoista pienempiä kuin nolla, mutta Y:lle voi asettaa alarajaksi nolla, jolloin tästä ongelmasta päästään. Liitteet kulkukuva ja tiheydet liittyvät tähän ja aukeavat bugs:ssa. Taulukossa 3 iät on jaettu neljään ryhmään kolmen vuoden jaksoissa. 1-3, 4-6 jne. Sekä näitä vastaavat tulokset. Huomataan ettei miinuksia ole keskiarvoissa, joskin luottamusväleistä niitä löytyy. Taulukon 2 estimaatit ovat vain puuttuville arvoille. Mediaanit ovat melko hyviä arvioita, mutta luottamusvälit leviävät kohtuuttoman isoiksi, johtuen liian suuresta määrästä puuttuvia havaintoja.
+We can compare the results with data and draw conclusions. The median estimates seem to have pretty ok values compared the the original data.
+=== Rat example  ===
+This model can be found from WinBUGS help page. Our example is based directly to this model.
-Taulukko 2. WinBUGSin tuloslistaus silakkadatalle yhden vuoden ikä-intervallilla. <br><br>
-<br>
-Taulukko 3. WinBUGSin tuloslistaus silakkadatalle kolmen vuoden ikä-intervallilla. <br>
+Rats: a normal hierarchical model
-<br>
+This example is taken from section 6 of Gelfand et al (1990), and concerns 30 young rats whose weights were measured weekly for five weeks. Part of the data is shown below, where Yij is the weight of the ith rat measured at age xj.
-.2. Koodit ja niiden ajaminen
-Liitteenä tulee koodit model1 , noodle_1, ja data ja data_1 joista data:ssa on 1-4 ikäluokat ja data_1 löytyy kaikille iän arvoille vastaava, sekä inits jossa alkuarvoja, mikäli niitä ei tahdota simuloida.
+Weights Y<sub>ij</sub> of rat i on day x<sub>j</sub>
-Kaikki koodit aukeavat WinBUGS ohjelmalla. Ja ovat vapaasti päivitettävissä. Ajot on tehty mallilla model1 ja datalla data ja data_1 ja todettu toimiviksi alkuarvoilla inits. Vastaavanlainen malli on esimerkeissä joka löytyy helpin alta examples vol1 Rats.
+{| cellspacing="1" cellpadding="1" border="1" style="width: 221px; height: 108px;"
+|-
+|
+| x<sub>j</sub>=8
+| 15
+| 22
+| 29
+| 36
+|
+|-
+| Rat 1
+| 151
+| 199
+| 246
+| 283
+| 320
+|
+|-
+| Rat 2
+| 145
+| 199
+| 149
+| 293
+| 354
+|
+|-
+| Rat 30
+| 153
+| 200
+| 244
+| 288
+| 324
+|
+|}
-rajan asettaminen tapahtuu laittamalla Y:n koodin eli jakauman ja arvojen perään symbolin I(0,)
+ A plot of the 30 growth curves suggests some evidence of downward curvature.
-Ajaminen:
+The model is essentially a random effects linear growth curve
-Mikäli ongelmia niin soita/tule kysymään. Tarvitset kolme ikkunaa auki. Modelin alta model specification, Inferencen alta samples ja modelin alta update, niin pääsee tekemään samat ajot kuin mitä on tässä tehty.
+Y<sub>ij</sub> ~ Normal(a<sub>i</sub> + b<sub>i</sub>(x<sub>j</sub> - x<sub>bar</sub>), T<sub>c</sub>)    a<sub>i</sub> ~ Normal(a<sub>c</sub>, T<sub>a</sub>)    b<sub>i</sub> ~ Normal(b<sub>c</sub>, T<sub>b</sub>)   where x<sub>bar</sub> = 22, and T represents the precision (1/variance) of a normal distribution. We note the absence of a parameter representing correlation between a<sub>i</sub> and b<sub>i</sub> unlike in Gelfand et al 1990. However, see the Birats example in Volume 2 which does explicitly model the covariance between a<sub>i</sub> and b<sub>i</sub>. For now, we standardise the x<sub>j</sub>'s around their mean to reduce dependence between a<sub>i</sub> and b<sub>i</sub> in their likelihood: in fact for the full balanced data, complete independence is achieved. (Note that, in general, prior independence does not force the posterior distributions to be independent).
-Koodit löytyy: N:\Huippuyksikko\Tutkimus\R55_PMvsDioxin\MeHgvsOmega-3\mallit\Bayes\Aiemmat ajot ja pähkäilyt
+a<sub>c</sub> , t<sub>a</sub>?, b<sub>c</sub> , T<sub>b</sub>?, T<sub>c</sub> are given independent "noninformative''priors, with two alternatives considered for T<sub>a</sub> and T<sub>b</sub>: prior 1 is uniform on the scale of the standard deviations s<sub>a</sub> = 1/sqrt(T<sub>a</sub>) and a<sub>b</sub> = 1/sqrt(T<sub>b</sub>), and prior 2 is a gamma(0.001, 0.001) on the precisions T<sub>a</sub> and T<sub>b</sub>. Interest particularly focuses on the intercept at zero time (birth), denoted a<sub>0</sub> = a<sub>c</sub> - v<sub>c </sub>x<sub>bar</sub>. ''
-<br>5. JOHTOPÄÄTÖKSET
+Graphical model for rats example (using prior 1):
-Bugs ohjelmistolla saatuja tuloksia voi verrata silakka-dataan, ja sen pohjalta tehdä päätelmiä.<br><br>
-<br>
+BUGS language for rats example:
-ESIMERKKI 1.
-Hierarkinen rotta esimerkki, joka löytyy winbugsin helpistä. Tähän malliin perustuu suoraan meidän käyttämät mallit.
-Rats: a normal hierarchical model
+Note the use of a very flat but conjugate prior for the population effects: a locally uniform prior could also have been used.
-This example is taken from section 6 of Gelfand et al (1990), and concerns 30 young rats whose weights were measured weekly for five weeks. Part of the data is shown below, where Yij is the weight of the ith rat measured at age xj.
+Data list(x = c(8.0, 15.0, 22.0, 29.0, 36.0), xbar = 22, N = 30, T = 5,   Y = structure(  .Data = c(151, 199, 246, 283, 320,  145, 199, 249, 293, 354,  147, 214, 263, 312, 328,  155, 200, 237, 272, 297,  135, 188, 230, 280, 323,  159, 210, 252, 298, 331,  141, 189, 231, 275, 305,  159, 201, 248, 297, 338,  177, 236, 285, 350, 376,  134, 182, 220, 260, 296,  160, 208, 261, 313, 352,  143, 188, 220, 273, 314,  154, 200, 244, 289, 325,  171, 221, 270, 326, 358,  163, 216, 242, 281, 312,  160, 207, 248, 288, 324,  142, 187, 234, 280, 316,  156, 203, 243, 283, 317,  157, 212, 259, 307, 336,  152, 203, 246, 286, 321,  154, 205, 253, 298, 334,  139, 190, 225, 267, 302,  146, 191, 229, 272, 302,  157, 211, 250, 285, 323,  132, 185, 237, 286, 331,  160, 207, 257, 303, 345,  169, 216, 261, 295, 333,  157, 205, 248, 289, 316,  137, 180, 219, 258, 291,  153, 200, 244, 286, 324),  .Dim = c(30,5)))
-Weights Yij of rat i on day xj<br> xj = 8 15 22 29 36 <br> __________________________________ <br> Rat 1 151 199 246 283 320<br> Rat 2 145 199 249 293 354<br> .......<br> Rat 30 153 200 244 286 324
+(Note: the response data (Y) for the rats example can also be found in the file ratsy.odc in rectangular format. The covariate data (X) can be found in S-Plus format in file ratsx.odc. To load data from each of these files, focus the window containing the open data file before clicking on "Data" from the "Model" menu.)
-<br>A plot of the 30 growth curves suggests some evidence of downward curvature.
+Inits1 list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250,   250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250),  beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,   6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6),
-The model is essentially a random effects linear growth curve
+alpha.c = 150, beta.c = 10,   tau.c = 1, sigma.alpha = 1, sigma.beta = 1)
-Yij ~ Normal(i + i(xj - xbar), c)<br> <br> i ~ Normal(c, )<br> <br> i ~ Normal(c, )<br> <br>where xbar = 22, and  represents the precision (1/variance) of a normal distribution. We note the absence of a parameter representing correlation between i and i unlike in Gelfand et al 1990. However, see the Birats example in Volume 2 which does explicitly model the covariance between i and i. For now, we standardise the xj's around their mean to reduce dependence between i and i in their likelihood: in fact for the full balanced data, complete independence is achieved. (Note that, in general, prior independence does not force the posterior distributions to be independent).
+Inits2 list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250,   250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250),  beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,   6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6),   alpha.c = 150, beta.c = 10,   tau.c = 1, tau.alpha = 1, tau.beta = 1)
-c , , c , , c are given independent "noninformative'' priors, with two alternatives considered for  and : prior 1 is uniform on the scale of the standard deviations  = 1/sqrt() and  = 1/sqrt(), and prior 2 is a gamma(0.001, 0.001) on the precisions  and . Interest particularly focuses on the intercept at zero time (birth), denoted 0 = c - c xbar.
+Results
-Graphical model for rats example (using prior 1):
+A 1000 update burn in followed by a further 10000 updates gave the parameter estimates:
+node mean sd MC error 2.5% median 97.5% start sample  alpha0 106.6 3.65 0.04151 99.43 106.5 113.9 1001 10000  beta.c 6.185 0.1102 0.001294 5.967 6.185 6.404 1001 10000  sigma 6.074 0.4673 0.007724 5.247 6.044 7.068 1001 10000
+Using Prior 2 (not recommended) has negligible impact
-BUGS language for rats example:
+These results may be compared with Figure 5 of Gelfand et al 1990 --- we note that the mean gradient of independent fitted straight lines is 6.19.
+Gelfand et al 1990 also consider the problem of missing data, and delete the last observation of cases 6-10, the last two from 11-20, the last 3 from 21-25 and the last 4 from 26-30. The appropriate data file is obtained by simply replacing data values by NA (see below). The model specification is unchanged, since the distinction between observed and unobserved quantities is made in the data file and not the model specification.
+click on one of the arrows to open the data for the missing value analysis
+Gelfand et al 1990 focus on the parameter estimates and the predictions for the final 4 observations on rat 26. These predictions are obtained automatically in BUGS by monitoring the relevant Y[ ] nodes. The following estimates were obtained:
-<br>Note the use of a very flat but conjugate prior for the population effects: a locally uniform prior could also have been used.
+We note that our estimate 6.58 of bc is substantially greater than that shown in Figure 6 of Gelfand et al 1990. However, plotting the growth curves indicates some curvature with steeper gradients at the beginning: the mean of the estimated gradients of the reduced data is 6.66, compared to 6.19 for the full data. Hence we are inclined to believe our analysis. The observed weights for rat 26 were 207, 257, 303 and 345, compared to our predictions of 204, 250, 295 and 341.
-Data list(x = c(8.0, 15.0, 22.0, 29.0, 36.0), xbar = 22, N = 30, T = 5, <br> Y = structure(<br> .Data = c(151, 199, 246, 283, 320,<br> 145, 199, 249, 293, 354,<br> 147, 214, 263, 312, 328,<br> 155, 200, 237, 272, 297,<br> 135, 188, 230, 280, 323,<br> 159, 210, 252, 298, 331,<br> 141, 189, 231, 275, 305,<br> 159, 201, 248, 297, 338,<br> 177, 236, 285, 350, 376,<br> 134, 182, 220, 260, 296,<br> 160, 208, 261, 313, 352,<br> 143, 188, 220, 273, 314,<br> 154, 200, 244, 289, 325,<br> 171, 221, 270, 326, 358,<br> 163, 216, 242, 281, 312,<br> 160, 207, 248, 288, 324,<br> 142, 187, 234, 280, 316,<br> 156, 203, 243, 283, 317,<br> 157, 212, 259, 307, 336,<br> 152, 203, 246, 286, 321,<br> 154, 205, 253, 298, 334,<br> 139, 190, 225, 267, 302,<br> 146, 191, 229, 272, 302,<br> 157, 211, 250, 285, 323,<br> 132, 185, 237, 286, 331,<br> 160, 207, 257, 303, 345,<br> 169, 216, 261, 295, 333,<br> 157, 205, 248, 289, 316,<br> 137, 180, 219, 258, 291,<br> 153, 200, 244, 286, 324),<br> .Dim = c(30,5)))
-(Note: the response data (Y) for the rats example can also be found in the file ratsy.odc in rectangular format. The covariate data (X) can be found in S-Plus format in file ratsx.odc. To load data from each of these files, focus the window containing the open data file before clicking on "Data" from the "Model" menu.)
-Inits1 list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, <br> 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250),<br> beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, <br> 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6),
+Table 4. Original data for herring
-<br> alpha.c = 150, beta.c = 10, <br> tau.c = 1, sigma.alpha = 1, sigma.beta = 1)
+{| {{prettytable}}
+|-
+| Age
+| Weight
+| Concentration
+|-
+| 2
+| 15
+| 1.96
+|-
+| 4
+| 23
+| 2.49
+|-
+| 6
+| 31
+| 6.58
+|-
+| 7
+| 43
+| 13.4
+|-
+| 11
+| 71.7
+| 17.2
+|-
+| 2
+| 17,6
+| 0.876
+|-
+| 3
+| 25.2
+| 2.22
+|-
+| 3
+| 34.1
+| 1.99
+|-
+| 5
+| 54
+| 1.5
+|-
+| 6
+| 137
+| 2.18
+|-
+| 2
+| 18
+| 2
+|-
+| 2
+| 24
+| 2.52
+|-
+| 6
+| 33
+| 8.23
+|-
+| 9
+| 52
+| 15.9
+|-
+| 11
+| 64
+| 16.6
+|-
+| 2
+| 16.3
+| 0.704
+|-
+| 4
+| 23.6
+| 2.84
+|-
+| 7
+| 34
+| 6.79
+|-
+| 9
+| 45
+| 14.5
+|-
+| 12
+| 65
+| 13.6
+|-
+| 2
+| 17
+| 2.29
+|-
+| 3
+| 24
+| 3.67
+|-
+| 7
+| 33
+| 10.9
+|-
+| 8
+| 42.9
+| 11.2
+|-
+| 9
+| 55
+| 17.7
+|-
+| 1
+| 15.3
+| 0.733
+|-
+| 3
+| 23.5
+| 2.35
+|-
+| 5
+| 30
+| 4.59
+|-
+| 7
+| 41
+| 6.59
+|-
+| 3
+| 15.8
+| 2.02
+|-
+| 6
+| 22
+| 4.6
+|-
+| 7
+| 29
+| 7.4
+|-
+| 7
+| 47
+| 5.03
+|-
+| 6
+| 144
+| 7.52
+|-
+| 1
+| 19
+| 1.13
+|-
+| 3
+| 25
+| 1.186
+|-
+| 5
+| 36
+| 5.7
+|-
+| 2
+| 13
+| 1.42
+|-
+| 5
+| 22
+| 3.25
+|-
+| 6
+| 32
+| 4.44
+|-
+| 8
+| 34
+| 6.58
+|-
+| 10
+| 106
+| 3.13
+|-
+| 1
+| 17
+| 0.768
+|-
+| 3
+| 24
+| 1.79
+|-
+| 2
+| 24
+| 0.782
+|-
+| 3
+| 48
+| 1.29
+|-
+| 7
+| 196
+| 2.31
+|-
+|
+|
+|
+|}
-<br>Inits2 list(alpha = c(250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, <br> 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250, 250),<br> beta = c(6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, <br> 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6), <br> alpha.c = 150, beta.c = 10, <br> tau.c = 1, tau.alpha = 1, tau.beta = 1)
-<br>Results
+==== Tutkittava asia Panulle: ====
-A 1000 update burn in followed by a further 10000 updates gave the parameter estimates:
+'''Summa PBDE pitoisuus'''
-node mean sd MC error 2.5% median 97.5% start sample<br> alpha0 106.6 3.65 0.04151 99.43 106.5 113.9 1001 10000<br> beta.c 6.185 0.1102 0.001294 5.967 6.185 6.404 1001 10000<br> sigma 6.074 0.4673 0.007724 5.247 6.044 7.068 1001 10000
+Luokitus ensisijaisesti rasvaprosentin mukaan (vaihtoehtoisesti ikä, paino tai pituus). Jaetaan neljään lohkoon (esim. rasvaprosentti 0-2, 2-4, 4-6, &gt;6)
-Using Prior 2 (not recommended) has negligible impact
+Kokeile eri kategoriointa (rasvaprosentti, ikä, paino, pituus) mikä parhaiten selittää PBDE -pitoisuuden muuttumista.
+Samaan tapaan kuin aiempi silakka-data. Priori-jakauma tulee kaikista PBDE-mittauksista ja rasvapidoisuuden täydentävä data tulee kustakin saman rasvapitoisuusluokan omaavista kaloista.
+Data kirjasta Hallikainen et al. 2004. Sivut 17- 36. Kotimaisten järvi- ja merikalan dioksiinien, furaanien, dioksiinien kaltaisten PCB –yhdisteiden ja polybromattujen difenyylieettereiden pitoiuudet. EU-kalat. Elintarvikeviraston julkaisuja. Helsinki 2004.
-These results may be compared with Figure 5 of Gelfand et al 1990 --- we note that the mean gradient of independent fitted straight lines is 6.19.
+==See also==
-Gelfand et al 1990 also consider the problem of missing data, and delete the last observation of cases 6-10, the last two from 11-20, the last 3 from 21-25 and the last 4 from 26-30. The appropriate data file is obtained by simply replacing data values by NA (see below). The model specification is unchanged, since the distinction between observed and unobserved quantities is made in the data file and not the model specification.
+* [[SHELF| SHELF package for R and MATCH online tool]]
+* [[:op_fi:Ven%C3%A4j%C3%A4n_vaalit_2011#Hierarkkinen_Bayes-malli_-_Hierarchial_Bayesian_model|Hierarchical Bayesian model of Russian duma election 2011]]
+** [[:op_fi:Tiedosto:RussianElections2011.ods|OpenBUGS model about election]] (does not work)
+* [http://www.opasnet.org/eractest/index.php/Sandbox#JAGS_test rjags test] (password required)
+* [http://www.opasnet.org/eractest/index.php/Sandbox#rbugs_test rbugs test] (password required) ([http://cran.r-project.org/web/packages/rjags/index.html rjags in CRAN])
+* [[:op_fi:Silakan riskiarvio|Benefit-risk assessment of Baltic herring]]
-click on one of the arrows to open the data for the missing value analysis
+== References  ==
-Gelfand et al 1990 focus on the parameter estimates and the predictions for the final 4 observations on rat 26. These predictions are obtained automatically in BUGS by monitoring the relevant Y[] nodes. The following estimates were obtained:
+<references/>
-We note that our estimate 6.58 of bc is substantially greater than that shown in Figure 6 of Gelfand et al 1990. However, plotting the growth curves indicates some curvature with steeper gradients at the beginning: the mean of the estimated gradients of the reduced data is 6.66, compared to 6.19 for the full data. Hence we are inclined to believe our analysis. The observed weights for rat 26 were 207, 257, 303 and 345, compared to our predictions of 204, 250, 295 and 341.<br><br>
+) Gelman Andrew, Carlin John B., Stern Hal S., and Rubin Donald B. Bayesian data analysis – second edition. Chapman &amp; Hall/CRC 2004.
+[[Category:Bayes]]