A statistical method to determine sample size to estimate characteristic value of soil parameters

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1 A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig characteristic values of soil parameters. The authors have proposed to determie sample size based both o Type I ad II errors i statistical hypothesis testig. I the previous study, however, oly idepedet samples are cosidered. I the preset study, spatially correlated samples are cosidered i the same framework. It is foud that the spatial autocorrelatio structure of soil parameters ad layer thickess are domiat factors i determiig the sample size. Itroductio I traditioal geotechical desig, how to determie soil parameter values is ot specifically defied. I the moder code draftig, however, this problem has attracted much attetio of code writers. I this study, we restrict the problem to the estimatio of a mea value of geotechical parameter whose variace (or COV), the spatial correlatio structure ad the distributio type are kow. The most commo procedure to determie the sample size i this case is to look for the cofidece iterval (so called Type I Error i the statistical hypothesis testig). For example, Lumb (974) used this idea for determiig umber of sample size based o Studet s t-test (the ormal distributio ca be used if the variace is kow, i.e. z-test). The sample size,, depeds o the t-value (or z-value), which i tur depeds o the sigificace level, α, set. The origiality of the procedure proposed i this study is that we look at ot oly Type I Error but also Type II Error. I order to propose the procedure, we have itroduced a ew parameter, which we call the distictio level,, which is give as: Departmet of Civil Egieerig, Gifu Uiversity, Yaagido -, Gifu JAPAN 50-93; Japa; hojo@cc.gifu-u.ac.jp 2 ditto; j38204@guedu.cc.gifu-u.ac.jp 3 Istitute of Techology, Shimizu Co., 3-4-7, Etchujima, Kohto-ku, Tokyo , Japa; makoto.suzuki@shimz.co.jp

2 ( alterative mea) ( ull mea) ( s.d. ) = () The detail explaatio of the procedure is give i Nagata(2003) ad Hojo ad Setiawa (2005). To be brief, we look at the miimum sample size ecessary to distiguish betwee the ull mea ad the alterative mea, which is give by. The ecessary sample size icreases as oe requires distictio betwee smaller. A procedure to determie ecessary sample size Hypothesis testig ad the distictio level: Hypothesis testig is a procedure of assessig whether sample data is cosistet or otherwise with statemets made about the populatio. A ull hypothesis (H 0 ) ad a alterative hypothesis (H ) eed to be itroduced first. A ull hypothesis (H 0 ) is a statistical hypothesis that is tested for possible rejectio uder the assumptio that it is true (i.e. observatios are the result of chace). The alterative hypothesis (H ) is the hypothesis agaist which the ull hypothesis is tested. I ay hypothesis testig there are two differet kids of errors that ca be committed, amely Type I ad Type II Errors: Type I Error: If a test leads to rejectio of the ull hypothesis ad hece whe the ull hypothesis is true the such a decisio is wrog. Probability of committig a Type I errors is α where α is termed sigificace level. Pr Reject H H 0 0 is true = α (2) Type II Error: If we miimize the chace of makig Type I error, we will ru the ievitable risk of acceptig ull hypothesis eve whe it is false. This is a Type II error, ad probability of committig this error is deoted by β where -β is termed power. Pr Accept H H 0 is true = β (3) I the framework of hypothesis testig, the Type I Error is oly cocered with H 0, where a cofidece iterval is fixed depedig o sigificace level, α. The cofidece iterval esures the estimated mea to be withi the iterval for probability -α. It must be recogized that oe caot make the probability of Type I Error arbitrarily small. Type I Error ca oly be made small at the expese of icreasig the probability of Type II Error. Power, -β represets the probability of rejectig the H 0 whe it is false. This represets the probability of the testig procedure correctly cocludig that H 0 is false whe H is true. To put this power testig i our problem, we have itroduced the distictio level, which is defied i Eq.(). By itroducig, we ca set appropriate H agaist H0 i a geeral ad parametric way. Sample size determiatio: If oe wats to distiguish the mea value to alterative hypothesis that the mea is σ smaller tha the mea assumed i H 0:µ = µ 0, H is give as H : µ µ 0 σ. This is so called oe sided test i the hypothesis testig.

3 The ecessary sample size to satisfy required α ad -β ca be determied by a table or figure calculated based o the followig Mote Carlo simulatio (MCS):. Select the ull hypothesis, alterative hypothesis, the sample size ad the umber of Mote Carlo Simulatio ru m. 2. Set sigificace level α 3. Geerate a series of samples ( x,, x ) followig a PDF assumig H 0 is true, i.e. ( ) 0, f x µ σ, ad calculate sample mea x obtaied as x = x j j= Repeat the procedure for m times ad geerate m x s. 4. Order x s from the lowest to the highest, ad redefie them as x i (i =,, m), ad Create the cofidece iterval [ xl α, + ), where xlα = lower boudary of the cofidece iterval for sigificace level α. 5. Geerate a sequece of radom variable ( x,, x ) followig a PDF assumig H is true, i.e. f( x µ 0 σσ, ), ad calculate sample mea x obtaied as x = j= x j. Repeat the procedure for m times ad geerate m x s. 6. Cout the relative umber of samples fall ito the iterval [ xl α, + ). Thus the power β is obtaied. By this procedure, figures ad tables ca be systematically created, which gives the relatioship betwee, α ad -β. A Example of such a figure is give i Fig. L = case. I this case, α is set to 0., whereas = 0. If is 0.8, α=β=0.9 is satisfied. A series of curves ad tables created i this way to obtai ecessary sample sizes for various coditios ca be see i Hojo ad Setiawa (2005).. β Figure. vs. -β for various L with α = 0. ad = 0 Ifluece of the spatial correlatio structure: So far, we have ot cosidered the spatial correlatio structure of the samples. I other words, it was implicitly assumed that all the samples are idepedet. However, this assumptio is far from the reality as far as soil samplig is cocered. The spatial correlatio of soil parameters are ofte described by a expoetial type auto-correlatio fuctio: 3

4 ρ( x) = exp x/ θ (4) where ρ is correlatio coefficiet, x is distace betwee two poits i the space, θ autocorrelatio distace. The loger the θ, stroger correlatio betwee a soil property for the same distace. The other importat factor that affects the problem is the thickess of a soil layer L. Sice the layer thickess is limited, the umber of samples oe ca take is limited assumig the sample iterval is the same. Two samples are geerated for differet autocorrelatio distace, θ, which have the same mea ad s.d., are illustrated i Fig.2. For the process with the shorter θ, the fluctuatio takes place withi the shorter scale. O the other had, the geerated samples take relatively similar values for loger θ process. It is uderstood from this figure that auto-correlatio distace, θ, ad layer thickess, L, are also importat factors i determiig ecessary sample size. I order to carry out parametric study to see the ifluece of these factors, the layer thickess is ormalized by the autocorrelatio distace as L =L/θ. L is termed ormalized layer thickess hereafter. The procedure to determie the ecessary sample size by MCS is essetially the same as the oe preseted i the previous sectio. Oly differece is that the correlatio betwee the adjacet samples is take ito accout i sample geeratio i the form of coditioal PDF. θ = θ = 0 Fig. 2. Examples of simulated radom processes with differet θ. Results of calculatio ad discussio Figures ad tables that are useful i determiig sample size for various coditios have bee systematically geerated i this study, of which oly part of the results are preseted here to highlight some importat features. The most importat factors i these simulatios are ormalized layer thickess, L, distiguish level,, ad sigificace level, α. We cosider it appropriate to set the power, -β, equal to -α to determie samples size. L are set to 0., 0.5,.0, 5.0, 0 ad ifiity (i.e. idepedet), whereas α is set to 0.05, 0.0 ad 0.5. I the oe sided hypothesis testig, settig α=0.5 is almost equivalet to set the lower boud of the cofidece iterval equal to mea oe s.d. is moved betwee 0.0 ad

5 β Figure 3(a). vs. -β with various, where α = 0. ad L = 0 β Figure 3(b). vs.-β with various, where α = 0. ad L =.0 The ifluece of L o ecessary sample size is featured i Fig., where α=0. ad =0 are fixed. It is clearly see from the figure that the idepedet case is most effective as far as sample size is cocered. For example, =0 is far sufficiet to esure -β be more tha 0.9 for =.0. Whe L is small, it is very difficult to reserve ecessary power, -β. The reaso is that if the ormalized layer thickess is short, samples oe ca get is ted to be biased ad do ot cotai sufficiet iformatio to estimate true mea of the populatio. The same fact ca be observed i Figs 3(a) ad (b), where powers are idicated for takig sample umber,, as a parameter. It is clearly see from the figures that sample size has cosiderable ifluece o power whe L is large like 0. However, it ifluece is almost ull whe L is small like.0. Part of the results of the calculatio is summarized i Table. Based o this table, oe ca quickly fid out the ecessary sample size for differet combiatios of a ad L for =.0 ad α=β coditio. Whe L is small, there are cases we caot estimate mea value oly from oe groud coditio (i.e. oe borig log) o matter how may samples oe gets from that oe log. 5

6 Table. Necessary sample size for estimatig mea value i case =.0 ad α=β. -α -β L = L = 0.0 L = 5.0 L =.0 85 % NP 90 % 7 NP NP 95 % 4 NP NP (ote) NP: Not possible. L= implies all the samples are idepedetly ad idetically distributed (i.i.d.). Coclusio It is preseted i this study that the assumptio of idepedet sample from a populatio, which is very stadard assumptio i the hypothesis testig, is ot appropriate whe applied to the soil sample size problem. The correlatio distace ad layer thickess have domiat ifluece especially the layer thickess is relatively short ad the autocorrelatio distace is relatively log. However, it should be clearly recogized that the problem pursued i this study is the estimatio of statistics of the whole populatio. I the cotext of geotechical egieerig, it is a problem to obtai a mea value of very large mass of soil whose COV ad autocorrelatio distace are roughly kow. For example, mea value of u-draied shear stregth of a port area. This problem ca be termed the ocoditioal estimatio by probabilistic termiologies. The situatio is very differet if oe wats to kow soil coditio uder a small structure, for example a house or a small buildig, which is goig to be built. I such case, the problem becomes the coditioal estimatio, ad the ucertaity at certai poit with measuremets reduce cosiderably. It is very importat to distiguish betwee o-coditioal estimatio problem ad coditioal estimatio problem i soil mechaics, ad oly sample size for ocoditioal estimatio problem is studied i the preset study. Our ext aim is to propose a procedure to determie ecessary sample size i the coditioal estimatio problems, which are cosidered to be more commo ad importat i geotechical egieerig. Refereces Hojo, Y ad B. Setiawa (2005), Appropriate sample size for determiig characteristic value of soil parameter by a statistical method, Proc. 50 th JGS symposium (submitted). Lumb, P. (974). Applicatio o f Statistics i Soil Mechaics. Soil Mechaic New Horizo, edited by Lee, I. K., pp. 44-0, Newess Butterworths, Lodo Nagata, Y. (2003). Determiatio of sample size (i Japaese). Asakura Syote, pp

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