A graphc comparson of the Feller and Delta ntervals for ratos of parameter estmates. Joe Hrschberg and Jenny Lye Economcs, Unversty of Melbourne. September 017 1
Introducton Feller s 1954 proposal for the use of an nverse test to construct confdence ntervals (Cs) for the rato of normally dstrbuted statstcs has been shown to be superor to the applcaton of the Delta method n several applcatons. In ths presentaton, we demonstrate how a smple graphc exposton can be generated to llustrate the relatonshp between the Delta and the Feller Cs. The advantage of the graphcal presentaton over the numerc result avalable n the feller Stata command (Coveney 004) s that t may ndcate how the level of sgnfcance can be changed to result n fnte upper and lower bounds. And to be used n cases where the Feller results are not defned.
We defne two normally dstrbuted random varables as: ˆ σ {[ ] } 1 σ 1 ˆ, γβ ~ N γβ,, Ω, where Ω= σ1 σ Where the quantty of nterest s the rato defned as γ θ= β ˆ Whch has been estmated by θ= ˆ γ β ˆ The objectve of ths analyss s to determne the confdence nterval for ˆθ. 3
The estmaton of the confdence nterval can be done va the applcaton of the Delta approxmaton method (.e. the use of the Stata nlcom command) or alternatvely they may employ the Feller method for the approxmaton of the confdence nterval. Many papers have compared these approxmatons and found that they concde for cases when the denomnator varable s βˆ estmated wth a low relatve varance (.e. tβ = σˆ s large (> 3)). However, n other cases the Feller has been shown to provde superor coverage (See Hrschberg and Lye 010) for comparsons). Note that Feller ntervals are not forced to be symmetrc as are the Delta ntervals. 4
The Inverse Test for the rato (Feller) 1 One method for determnng the value of the rato s to rewrte ˆ the formula ˆ γ θ= as γˆ θβ ˆˆ= 0. β ˆ Thus we could use the values of γβ ˆ, ˆ and try dfferent values of θ=θ to search for the value of θ where γ θβ= ˆ *ˆ 0 s true then conclude that ˆθ=θ. * 1 See our paper Lye and Hrschberg (01) for more examples of smlar nverse tests. 5
The beneft of ths approach s that the lnear functon γ θβ ˆ ˆ of normally dstrbuted random varables s normally dstrbuted. ˆ Whle the rato of normally dstrbuted random varables ˆ γ θ=, s β ˆ dstrbuted va a Cauchy dstrbuton that does not have fnte moments. However, ths does not preclude the constructon of probablty statements concernng ˆθ. 6
by: We can test our canddate values of θ va the hypothess defned H H 0 1 : γ θβ= ˆ ˆ 0 : γ θβ ˆ ˆ 0 Under H 0 we can construct a test statstc defned by the rato of the lnear combnaton evaluated at θ : ( ) Var ( ) T ˆ = γ θβ ˆ ˆ γ θβ ˆ ˆ Would be dstrbuted as the square of a standard normal or a Chsquare wth 1 degree of freedom. 7
We nvert the problem by determnng the value(s) of θ where the test statstc s equal to the value of the square of standard z α : normal for the level of α ( ) ( ˆ ) Var ( ˆ ) z α = γ θβˆ γ θβ ˆ as the lmts of the confdence nterval. We desgnate these lmts as θ 0. Usng the estmate of ˆ σˆ1 σˆ 1 Ω=, ths s equvalent to the quadratc n θ 0: σˆ1 σˆ ( ˆ ) ( ) ( ) ˆ z ˆ α θ ˆ ˆ 0 + ˆ1zα θ 0 + ˆ1 zα β σ σ γβ γ σ = 0 The two roots of ths quadratc defne the potental solutons for θ 0 and provde the lmts of the confdence nterval. 8
The soluton to ths quadratc are defned by: θ θ = {, } U L ( ˆ ˆ ˆ) 1z σ βγ ± α α ( βˆ σˆ ) z 0 0 z βˆ σ ˆ +γˆ σ ˆ +σˆ 1 1 σˆ σˆ 1 1 α z α βγσ ˆ ˆ ˆ Computng the soluton to ths quadratc equaton s wdely avalable (.e. the polyval command). However, the quadratc may not have real roots. Furthermore, the numbers generated my not provde much ntuton as to how the nterval may change wth dfferng values of α. z α 9
The Lne-plot Verson of the Feller The alternatve approach to computng the roots of the mpled quadratc equaton s the lne-plot approach. Ths approach returns to the formulaton: ( ) γ θβ ˆ ˆ 0 z α = ( ) Var γ θβ ˆ ˆ 0 After takng the square-root of both sdes we have: ˆ γ θβ ˆ ± Var γ θβ ˆ = 0 ( ˆ ) ( ˆ ) z α 0 0 Where ( γ θβ ˆ ) ± Var ( ˆ ) z α of the lnear combnaton ( ˆ ) γ θβ ˆ defnes the confdence nterval γ θβ ˆ for dfferent θs. 10
Thus by plottng the lne defned by γ θβ ˆ ˆ over dfferent values ˆ of θ around the value of the estmated rato ˆ γ θ= we can fnd the β ˆ confdence bounds { θ } 0U, θ0l at the values of confdence nterval of the lne defned by: cuts the zero-axs lne. ˆ ( γ θβ ˆ ) ± Var ( γ θβ ˆ ) z α ˆ θ where the 11
An Example of the rato of the means of a common varable. The means of dfferent sze samples of a common varable s a specal case where the covarance between the means ( σ 1 ) s set to zero and the varances are equal ( σ 1 =σ ). Here we use the property of a least squares regresson wth two dummy varables and no ntercept. Y = γ D +β D +ε t 1t t t Ths specfcaton wll estmate the means of Y for two groups when we assume the varance of Y s the same for both groups. 1
The estmate of the rato of the means s defned by the rato of the ˆ two estmated parameters ˆ γ θ=. β ˆ By usng a regresson we can employ the margns command, as well as the nlcom command to provde a comparson to the Delta method. The Stata code below reads the smulated blood pressure data, creates two dummes by the varable sex and computes the two means va a regresson. 13
Ths data has an ndcator for sex and we frst create two dummes for each sex. Then we generate a new varable _xx_ that wll only be used here. The regresson estmates the two means (note that _xx_ s not used n the regresson but we need t for the margns command also we need to drop the constant n ths case) tabulate sex, generate(sex) gen _xx_ = 0 label var _xx_ "Rato" reg bp Sex1 Sex _xx_, noconstant See program Feller_for_nd_corr_means.do for detals.. 14
We frst use the nlcom command to determne the Delta method 95% confdence nterval of the rato. nlcom _b[sex1] / _b[sex], level(95) Ths results n: bp Coef. Std. Err. z P> z [95% Conf. Interval] _nl_1.5301097.46018 1.15 0.51 -.3757891 1.436009 Thus the rato s.5301097 and the Delta CI s -.375789 to 1.436009 a symmetrc nterval. 15
To generate the lne-plot verson of the Feller we can use the margns command to estmate the value of the lnear combnaton γ θβ ˆ ˆ where _xx_ s the value of θ along wth the 95% confdence nterval. margns, expresson (_b[sex1]-_b[sex]*_xx_) at(_xx_=(-1(.1)5)) level(95) Note that we have set the lmts of the expresson to be between - 1 and 5 as the lmtng values over whch we evaluate the possble values of the confdence nterval. 16
Next we use the margnsplot command to plot the lnear combnaton along wth the 95% nterval. margnsplot, ylne(0) xlne(0) recast(lne) recastc(rlne) name(rato_nd, replace) ttle(feller CI- Rato of Independent Means) We add the zero-zero axs to evaluate our hypothess. 17
Feller CI - Rato of Independent Means _b[sex1]-_b[sex]*_xx_ -4000-3000 -000-1000 0 1000-0 4 6 Rato Ths s the Stata produced graph. 18
( γ θβ ˆ ˆ) + z α Var ( γ θβ ˆ ˆ) γ θβ ˆ ˆ ( γ θβ ˆ ˆ) z α Var ( γ θβ ˆ ˆ) θ= ˆ γˆ β ˆ = θ The tems plotted. 19
Delta 95% CI Feller 95% CI The dentfcaton of the 95% Feller and Delta CIs. 0
Alternatvely, the 99% Delta nterval s lsted as: nlcom _b[sex1] / _b[sex], level(99) wll always result n a fnte symmetrc CI: bp Coef. Std. Err. z P> z [99% Conf. Interval] _nl_1.5301097.46018 1.15 0.51 -.660443 1.70663 The results of the feller command however: feller bp, by(sex) level(99) results n: Quotent not statstcally sgnfcantly dfferent from zero at the 99 level. Confdence nterval not estmable 1
The equvalent plot however does provde more nformaton: _b[sex1]-_b[sex]*_xx_ -4000-000 0 000 4000 Feller CI - Rato of Independent Means Feller lower bound -4-0 4 6 Rato From ths plot we can observe a fnte lower bound wth an nfnte upper bound
An Example of the rato of the means from the same number of observatons. Ths example data contans data on observatons that are desgnated as before and after for the same ndvdual. We can use a seemngly unrelated regresson whch relaxes the assumptons ( σ 1 = 0, and σ 1 =σ ). However, t does requre that the number of observatons be equal. To use ths same example data as above, we frst need to reshape the data from a long format to a wde format usng the reshape command. reshape wde bp, (patent) j(when) 3
Then we estmate two equatons usng a seemngly unrelated regressons (SUR) specfcaton wth the sureg command: sureg (bp1 _xx_) (bp _xx_ ), corr Agan, we nclude the _xx_ varable thus we are nterested n the constants n equaton. Note that although the SUR procedure s a form of GLS the estmates n ths case are the same as OLS because the regressors are the same n both equatons, however the resultng covarance between the ntercepts wll be estmated as non-zero and the errors for each equaton wll be used to estmate separate varances. 4
The Delta nterval can be determned for ths rato can be estmated va the nlcom command where we dentfy the estmates of the constants from each equaton. nlcom [bp1]_b[_cons] / [bp]_b[_cons] Resultng n Coef. Std. Err. z P> z [95% Conf. Interval] _nl_1.686776.541614 1.7 0.05 -.3747645 1.748317 The rato s estmated as.686776 and the CI s approxmated as -.374 to 1.748. 5
As before we now use the margns and margnsplot commands to estmate and plot the confdence nterval for the lnear combnaton. The code for these steps are: margns, expresson ([bp1]_b[_cons]- [bp]_b[_cons]*_xx_) at(_xx_=(-1(.1)5)) level(95) margnsplot, ylne(0) xlne(0) recast(lne) recastc(rlne) name(rato_corr, replace) ttle(feller CI for Rato of Correlated Means) 6
Feller CI - Rato of Correlated Means [bp1]_b[_cons]-[bp]_b[_cons]*_xx_ -4000-3000 -000-1000 0 1000 Delta CI Feller CI - 0 4 6 Rato The comparson of the CIs for the correlated means. 7
The turnng pont of a quadratc. A common case of the rato of parameter estmates would be where one has ft a quadratc specfcaton and would lke to determne the locaton of the turnng pont. For example, the typcal specfcaton would be of the form: Y =α+γ X +β X +ε t t t t The turnng pont occurs when the margnal effect of the X varable s zero: Yt * γ =γ+ β Xt, thus X = X β t 8
* Where X s the value of X where the sgn of the margnal effect reverses n other words where the turnng pont occurs. For example, usng the Stata example data set auto we can estmate a quadratc between a car s prce and the weght of the car here we only consder the domestc cars n the data. 3 3 See Turnng_pont.do for the full set of code for ths example. 9
Frst we estmate a quadratc equaton where weght s the square of weght: reg prce weght weght Source SS df MS Number of obs = 5 F(, 49) = 39.08 Model 300678877 150339439 Prob > F = 0.0000 Resdual 18851593 49 384763.74 R-squared = 0.6146 Adj R-squared = 0.5989 Total 489194801 51 959054.9 Root MSE = 1961.4 prce Coef. Std. Err. t P> t [95% Conf. Interval] weght -9.84143.851178-3.45 0.001-15.57108-4.111766 weght.0019851.0004367 4.55 0.000.0011075.00866 _cons 15933.98 4555.578 3.50 0.001 6779.08 5088.76 30
The ftted regresson lne and data are shown below: 0 5,000 10,000 15,000,000 3,000 4,000 5,000 Weght (lbs.) Prce to weght relatonshp 31
To fnd the turnng pont we can use the followng code: nlcom (_b[weght] /( -*_b[weght]) ), level(95) Where the nlcom command provdes the Delta estmaton of the CIs for the turnng pont as: prce Coef. Std. Err. z P> z [95% Conf. Interval] _nl_1 478.865 193.4764 1.81 0.000 099.658 858.07 The turnng pont s estmated at,478.865 pounds and the approxmate 95% CI s,099.658 to,858.07. 3
Alternatvely, we can plot the margnal effect of the change n weght as t nfluences the prce. Usng the Stata code: margns, expresson (_b[weght] + *_b[weght]*weght) at (weght = (1500(00)3000)) level(95) margnsplot, ylne(0) recast(lne) recastc(rlne) name(turnng_pont, replace) ttle(feller CI - Turnng pont of quadratc ) 33
Feller CI - Turnng pont of quadratc _b[weght] + *_b[weght]*weght -8-6 -4-0 Feller CI Delta CI 1,500 1,700 1,900,100,300,500,700,900 Weght (lbs.) Turnng Pont Note that the Feller CI s asymmetrc and has a lower lmt that s less than the Delta nterval. (see Lye and Hrschberg 01 for more examples). 34
An Example of the 50% dose and Wllngness to Pay. The 50% dose problem was the orgnal (1944) applcaton that Feller consdered when he frst proposed ths method. In that case, he was nterested n the dose of a medcaton/poson that would result n a greater than 50% chance that the subject would lve/de. In the wllngness to pay case the varable appled s the prce pad (e.g. Hole 007). How much would someone pay for a product/servce? Thus, the data would be a dchotomous response to a contnuous stmulus-drug-prce. 35
Typcally, ths problem s modelled by a probablty of occurrence usng a logt or probt estmator. PY ( = 1) =α+γ X + β W +ε t t t t = 1 Where X s the varable dose/prce and the Ws are other covarates. The queston of the 50% dose s posed as: What value of X (desgnated as X * ) makes ths relatonshp k true? ½ k * =α+γ X + β = 1 W t 36
Solvng for X * we have the rato gven by: X * = ½ k α+ βw = 1 γ t The rato can be evaluated at specfc values of W for cases of nterest or at the means. 37
As an example, we use data on the survvors of the Ttanc dsaster were the dose/prce s the fare pad for the trp (see Frey et al 010). We ft the model to the data wth covarates for sex and age. 4 probt survve fare age female Whch results n: Probt regresson Number of obs = 1,033 LR ch(3) = 345.4 Prob > ch = 0.0000 Log lkelhood = -55.5508 Pseudo R = 0.473 survved Coef. Std. Err. z P> z [95% Conf. Interval] fare.0059587.0010804 5.5 0.000.0038411.0080763 age -.00655.003111 -.10 0.036 -.01669 -.0004348 female 1.440331.091084 15.81 0.000 1.61809 1.61885 _cons -.815731.107458-7.59 0.000-1.0584 -.60473 4 See the program Wllngness_to_pay.do for the full code for ths example. 38
Usng the Delta method for the 50% fare for the average age and assumng 50% gender splt: nlcom (-(_b[_cons]+ _b[age]*m_age + _b[female] *.5) / _b[fare] ), level(95) Results n the estmated CI on the 50% fare of: survved Coef. Std. Err. z P> z [95% Conf. Interval] _nl_1 48.67789 8.09075 6.0 0.000 3.81771 64.53806 Thus, we fnd a fare of 48.7 pounds wth an approxmate 95% nterval of (3.8 to 64.54). 39
To estmate the Feller nterval we use the margns command to predct the probablty of survval: margns, expresson ( normal( _b[_cons]+_b[fare]*fare + _b[age]*age + _b[female] * female ) ) at(age (mean) female =.5 fare=(0(5)100)) level(95) Followed by the margnsplot command: margnsplot, ylne(.5) recast(lne) recastc(rlne) 40
Feller CI - lmt of 50% dose Probablty of survval.3.4.5.6.7 50% Fare Feller CI Delta CI 0 5 10 15 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Fare pad n pounds In ths case the Delta and Feller ntervals concde qute closely. 41
Conclusons In addton to the examples dscussed here we have also added the routne used our 017 paper that estmates the ndrect leastsquares estmate for the just dentfed IV estmator whch can be shown to be the rato of parameters estmated from two seemngly unrelated regressons. 5 The types of code for these cases vares dependng on the best way to proceed. Usng the margns and the margnsplot command s a very compact method for generatng these plots. The lmts of the margns may need some adjustment after the frst attempt so that the bounds are plotted wthn the plot wndow. In addton, the level opton allows the constructon of CIs wth dfferent values of α. 5 See the program Indrect_Least_squares.do for ths example. 4
All programs/data used n ths talk are avalable at: https://www.onlne.fbe.unmelb.edu.au/t_drve/stata_examples do fles Feller_for_nd_corr.do Indrect_Least_Squares.do Turnng_pont.do Wllngness_to_pay.do Data bplong_example.dta maketable4.dta auto_domestc.dta ttanc1.dta 43
References Coveney, Joseph, (004), FIELLER: Stata module to calculate confdence ntervals of quotents of normal varate, downloaded 5/9/17, from : https://deas.repec.org/c/boc/bocode/s448101.html. Feller, E. C., (1944), A Fundamental Formula n the Statstcs of Bologcal Assay and Some Applcatons, Quarterly Journal of Pharmacy and Pharmacology, 17, 117-13.,(1954), Some Problems n Interval Estmaton, Journal of the Royal Statstcal Socety, Ser. B, 16, 174 185. Frey, B. S., D. A. Savage, and B. Torgler, 010, Interacton of natural survval nstncts and nternalzed socal norms explorng the Ttanc and Lustana dsasters, Proceedngs of the Natonal Academy of Scence, 107, 486-4865. Hole, A. R., 007, A comparson of approaches to estmatng confdence ntervals for wllngness to pay measures, Health Economcs, 16, 87-840. Hrschberg, Joseph G. and Jenny N. Lye, 010, A Geometrc Comparson of the Delta and Feller Confdence Intervals, The Amercan Statstcan, 64, 34-41., 017, Invertng the Indrect - The Ellpse and the Boomerang: Vsualsng the Confdence Intervals of the Structural Coeffcent from Two-Stage Least Squares, Journal of Econometrcs, 199, 173-183. Lye, Jenny N. and Joseph G. Hrschberg, 010, Two Geometrc Representatons of Confdence Intervals for Ratos of Lnear Combnatons of Regresson Parameters: An Applcaton to the NAIRU, Economcs Letters, 108, 73-76., 01, Inverse Test Confdence Intervals for Turnng Ponts: A Demonstraton wth Hgher Order Polynomals, Dek Terrell and Danel Mllmet (ed.) 30 th Annversary Edton (Advances n Econometrcs, Volume 30) Emerald Group Publshng Lmted, pp 59-95. 44