Analysis of trends

[Reza Hosseini]

Hello everyone,

My question is how to perform an analysis of trends during a certain period of time to see if the percentages of a desired outcome goes up or down significantly. Simply put, I have an outcome (as a percentage) for each year (10 years in total). The percentages go up and down and it's not easy to just look at the line graph and make a conclusion whether it went up or down during the 10 year period. Can anyone please guide me on how to perform such analysis? Previously I used ptrend command and weighted least square regression but the problem with those is that both methods "weigh" the percentages while I just want to see if the raw percentages go up or down during my study period.

Something like this:

year outcome
1991 .32
1992 .23
1993 .43
1994 .69
1995 .42
1996 .56
1997 .61
1998 .65
1999 .67
2000 .54


Thank you!

[Joseph Coveney]

Well, here's my try. Look at the first ("linear") component of the orthogonal polynomial contrasts shown with the contrast postestimation command below.

You would likely have a more powerful test if you could use the individual binomial values or at least the counts each year, if such data are available.

.ÿversionÿ14.1

.ÿ
.ÿclearÿ*

.ÿsetÿmoreÿoff

.ÿ
.ÿinputÿintÿyearÿdoubleÿoutcome

ÿÿÿÿÿÿÿÿÿyearÿÿÿÿÿoutcome
ÿÿ1.ÿ1991ÿ.32
ÿÿ2.ÿ1992ÿ.23
ÿÿ3.ÿ1993ÿ.43
ÿÿ4.ÿ1994ÿ.69
ÿÿ5.ÿ1995ÿ.42
ÿÿ6.ÿ1996ÿ.56
ÿÿ7.ÿ1997ÿ.61
ÿÿ8.ÿ1998ÿ.65
ÿÿ9.ÿ1999ÿ.67
ÿ10.ÿ2000ÿ.54
ÿ11.ÿend

.ÿ
.ÿgenerateÿdoubleÿsdÿ=ÿsqrt(outcomeÿ*ÿ(1ÿ-ÿoutcome))

.ÿ
.ÿvwlsÿoutcomeÿi.year,ÿsd(sd)

Variance-weightedÿleast-squaresÿregressionÿÿÿÿÿÿNumberÿofÿobsÿÿÿÿÿ=ÿÿÿÿÿÿÿÿÿ10
Goodness-of-fitÿchi2(0)ÿÿÿÿ=ÿÿÿÿÿÿÿ.ÿÿÿÿÿÿÿÿÿÿÿÿModelÿchi2(9)ÿÿÿÿÿ=ÿÿÿÿÿÿÿ1.08
Probÿ>ÿchi2ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ=ÿÿÿÿÿÿÿ.ÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿÿÿÿÿ=ÿÿÿÿÿ0.9992
------------------------------------------------------------------------------
ÿÿÿÿÿoutcomeÿ|ÿÿÿÿÿÿCoef.ÿÿÿStd.ÿErr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿConf.ÿInterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿyearÿ|
ÿÿÿÿÿÿÿ1992ÿÿ|ÿÿÿÿÿÿÿ-.09ÿÿÿ.6282515ÿÿÿÿ-0.14ÿÿÿ0.886ÿÿÿÿÿ-1.32135ÿÿÿÿÿ1.14135
ÿÿÿÿÿÿÿ1993ÿÿ|ÿÿÿÿÿÿÿÿ.11ÿÿÿ.6802206ÿÿÿÿÿ0.16ÿÿÿ0.872ÿÿÿÿ-1.223208ÿÿÿÿ1.443208
ÿÿÿÿÿÿÿ1994ÿÿ|ÿÿÿÿÿÿÿÿ.37ÿÿÿ.6568866ÿÿÿÿÿ0.56ÿÿÿ0.573ÿÿÿÿ-.9174741ÿÿÿÿ1.657474
ÿÿÿÿÿÿÿ1995ÿÿ|ÿÿÿÿÿÿÿÿÿ.1ÿÿÿ.6791171ÿÿÿÿÿ0.15ÿÿÿ0.883ÿÿÿÿ-1.231045ÿÿÿÿ1.431045
ÿÿÿÿÿÿÿ1996ÿÿ|ÿÿÿÿÿÿÿÿ.24ÿÿÿ.6811755ÿÿÿÿÿ0.35ÿÿÿ0.725ÿÿÿÿ-1.095079ÿÿÿÿ1.575079
ÿÿÿÿÿÿÿ1997ÿÿ|ÿÿÿÿÿÿÿÿ.29ÿÿÿ.6749074ÿÿÿÿÿ0.43ÿÿÿ0.667ÿÿÿÿ-1.032794ÿÿÿÿ1.612794
ÿÿÿÿÿÿÿ1998ÿÿ|ÿÿÿÿÿÿÿÿ.33ÿÿÿ.6671582ÿÿÿÿÿ0.49ÿÿÿ0.621ÿÿÿÿÿ-.977606ÿÿÿÿ1.637606
ÿÿÿÿÿÿÿ1999ÿÿ|ÿÿÿÿÿÿÿÿ.35ÿÿÿ.6623443ÿÿÿÿÿ0.53ÿÿÿ0.597ÿÿÿÿÿ-.948171ÿÿÿÿ1.648171
ÿÿÿÿÿÿÿ2000ÿÿ|ÿÿÿÿÿÿÿÿ.22ÿÿÿ.6826419ÿÿÿÿÿ0.32ÿÿÿ0.747ÿÿÿÿ-1.117954ÿÿÿÿ1.557954
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿ.32ÿÿÿ.4664762ÿÿÿÿÿ0.69ÿÿÿ0.493ÿÿÿÿ-.5942765ÿÿÿÿ1.234276
------------------------------------------------------------------------------

.ÿcontrastÿp.year,ÿpveffectsÿnowald

Contrastsÿofÿmarginalÿlinearÿpredictions

Marginsÿÿÿÿÿÿ:ÿasbalanced

-----------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿContrastÿÿÿStd.ÿErr.ÿÿÿÿÿÿzÿÿÿÿP>|z|
-------------+---------------------------------------
ÿÿÿÿÿÿÿÿyearÿ|
ÿÿÿ(linear)ÿÿ|ÿÿÿ.1054911ÿÿÿ.1493714ÿÿÿÿÿ0.71ÿÿÿ0.480
(quadratic)ÿÿ|ÿÿ-.0533967ÿÿÿ.1526225ÿÿÿÿ-0.35ÿÿÿ0.726
ÿÿÿÿ(cubic)ÿÿ|ÿÿ-.0130413ÿÿÿ.1520545ÿÿÿÿ-0.09ÿÿÿ0.932
ÿÿ(quartic)ÿÿ|ÿÿÿ-.006741ÿÿÿ.1499558ÿÿÿÿ-0.04ÿÿÿ0.964
ÿÿ(quintic)ÿÿ|ÿÿ-.0527641ÿÿÿ.1465866ÿÿÿÿ-0.36ÿÿÿ0.719
ÿÿÿ(sextic)ÿÿ|ÿÿÿ.0423435ÿÿÿ.1493729ÿÿÿÿÿ0.28ÿÿÿ0.777
ÿÿÿ(septic)ÿÿ|ÿÿ-.0078873ÿÿÿ.1520726ÿÿÿÿ-0.05ÿÿÿ0.959
ÿÿÿÿ(octic)ÿÿ|ÿÿ-.0385536ÿÿÿ.1518268ÿÿÿÿ-0.25ÿÿÿ0.800
ÿÿÿÿ(nonic)ÿÿ|ÿÿÿ.0409304ÿÿÿ.1545442ÿÿÿÿÿ0.26ÿÿÿ0.791
-----------------------------------------------------

.ÿ
.ÿ//ÿConfirmationÿthatÿtheÿregressionÿmodelÿfitsÿtheÿ"rawÿpercentages"
.ÿpredictÿdoubleÿxb,ÿxb

.ÿgraphÿtwowayÿscatterÿoutcomeÿyear,ÿmcolor(black)ÿmsize(small)ÿ||ÿ///
>ÿÿÿÿÿÿÿÿÿlineÿxbÿyear,ÿsortÿlcolor(black)ÿ///
>ÿÿÿÿÿÿÿÿÿylabel(ÿ,ÿangle(horizontal)ÿnogrid)ÿlegend(off)

.ÿ
.ÿ//ÿNoÿweighting,ÿonlyÿrankÿtransformation
.ÿnptrendÿoutcome,ÿby(year)

ÿÿÿÿÿÿÿÿÿÿyearÿÿÿÿÿÿÿÿscoreÿÿÿÿÿÿÿÿÿÿobsÿÿÿsumÿofÿranksÿÿ
ÿÿÿÿÿÿÿÿÿÿ1991ÿÿÿÿÿÿÿÿÿ1991ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ2ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1992ÿÿÿÿÿÿÿÿÿ1992ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1993ÿÿÿÿÿÿÿÿÿ1993ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ4ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1994ÿÿÿÿÿÿÿÿÿ1994ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿ10ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1995ÿÿÿÿÿÿÿÿÿ1995ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ3ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1996ÿÿÿÿÿÿÿÿÿ1996ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ6ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1997ÿÿÿÿÿÿÿÿÿ1997ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ7ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1998ÿÿÿÿÿÿÿÿÿ1998ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ8ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1999ÿÿÿÿÿÿÿÿÿ1999ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ9ÿÿ
ÿÿÿÿÿÿÿÿÿÿ2000ÿÿÿÿÿÿÿÿÿ2000ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ5ÿÿ

ÿÿÿÿÿÿÿÿÿÿzÿÿ=ÿÿ1.77
ÿÿProbÿ>ÿ|z|ÿ=ÿ0.076

.ÿ
.ÿexit

endÿofÿdo-file


.

Also, I'm not sure what you have against Stata commands (user-written ptrend or official nptrend) specifically written for tests of trend.

[Reza Hosseini]

Well, here's my try. Look at the first ("linear") component of the orthogonal polynomial contrasts shown with the contrast postestimation command below.

You would likely have a more powerful test if you could use the individual binomial values or at least the counts each year, if such data are available.

.ÿversionÿ14.1

.ÿ
.ÿclearÿ*

.ÿsetÿmoreÿoff

.ÿ
.ÿinputÿintÿyearÿdoubleÿoutcome

ÿÿÿÿÿÿÿÿÿyearÿÿÿÿÿoutcome
ÿÿ1.ÿ1991ÿ.32
ÿÿ2.ÿ1992ÿ.23
ÿÿ3.ÿ1993ÿ.43
ÿÿ4.ÿ1994ÿ.69
ÿÿ5.ÿ1995ÿ.42
ÿÿ6.ÿ1996ÿ.56
ÿÿ7.ÿ1997ÿ.61
ÿÿ8.ÿ1998ÿ.65
ÿÿ9.ÿ1999ÿ.67
ÿ10.ÿ2000ÿ.54
ÿ11.ÿend

.ÿ
.ÿgenerateÿdoubleÿsdÿ=ÿsqrt(outcomeÿ*ÿ(1ÿ-ÿoutcome))

.ÿ
.ÿvwlsÿoutcomeÿi.year,ÿsd(sd)

Variance-weightedÿleast-squaresÿregressionÿÿÿÿÿÿNumberÿofÿobsÿÿÿÿÿ=ÿÿÿÿÿÿÿÿÿ10
Goodness-of-fitÿchi2(0)ÿÿÿÿ=ÿÿÿÿÿÿÿ.ÿÿÿÿÿÿÿÿÿÿÿÿModelÿchi2(9)ÿÿÿÿÿ=ÿÿÿÿÿÿÿ1.08
Probÿ>ÿchi2ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ=ÿÿÿÿÿÿÿ.ÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿÿÿÿÿ=ÿÿÿÿÿ0.9992
------------------------------------------------------------------------------
ÿÿÿÿÿoutcomeÿ|ÿÿÿÿÿÿCoef.ÿÿÿStd.ÿErr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿConf.ÿInterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿyearÿ|
ÿÿÿÿÿÿÿ1992ÿÿ|ÿÿÿÿÿÿÿ-.09ÿÿÿ.6282515ÿÿÿÿ-0.14ÿÿÿ0.886ÿÿÿÿÿ-1.32135ÿÿÿÿÿ1.14135
ÿÿÿÿÿÿÿ1993ÿÿ|ÿÿÿÿÿÿÿÿ.11ÿÿÿ.6802206ÿÿÿÿÿ0.16ÿÿÿ0.872ÿÿÿÿ-1.223208ÿÿÿÿ1.443208
ÿÿÿÿÿÿÿ1994ÿÿ|ÿÿÿÿÿÿÿÿ.37ÿÿÿ.6568866ÿÿÿÿÿ0.56ÿÿÿ0.573ÿÿÿÿ-.9174741ÿÿÿÿ1.657474
ÿÿÿÿÿÿÿ1995ÿÿ|ÿÿÿÿÿÿÿÿÿ.1ÿÿÿ.6791171ÿÿÿÿÿ0.15ÿÿÿ0.883ÿÿÿÿ-1.231045ÿÿÿÿ1.431045
ÿÿÿÿÿÿÿ1996ÿÿ|ÿÿÿÿÿÿÿÿ.24ÿÿÿ.6811755ÿÿÿÿÿ0.35ÿÿÿ0.725ÿÿÿÿ-1.095079ÿÿÿÿ1.575079
ÿÿÿÿÿÿÿ1997ÿÿ|ÿÿÿÿÿÿÿÿ.29ÿÿÿ.6749074ÿÿÿÿÿ0.43ÿÿÿ0.667ÿÿÿÿ-1.032794ÿÿÿÿ1.612794
ÿÿÿÿÿÿÿ1998ÿÿ|ÿÿÿÿÿÿÿÿ.33ÿÿÿ.6671582ÿÿÿÿÿ0.49ÿÿÿ0.621ÿÿÿÿÿ-.977606ÿÿÿÿ1.637606
ÿÿÿÿÿÿÿ1999ÿÿ|ÿÿÿÿÿÿÿÿ.35ÿÿÿ.6623443ÿÿÿÿÿ0.53ÿÿÿ0.597ÿÿÿÿÿ-.948171ÿÿÿÿ1.648171
ÿÿÿÿÿÿÿ2000ÿÿ|ÿÿÿÿÿÿÿÿ.22ÿÿÿ.6826419ÿÿÿÿÿ0.32ÿÿÿ0.747ÿÿÿÿ-1.117954ÿÿÿÿ1.557954
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿ_consÿ|ÿÿÿÿÿÿÿÿ.32ÿÿÿ.4664762ÿÿÿÿÿ0.69ÿÿÿ0.493ÿÿÿÿ-.5942765ÿÿÿÿ1.234276
------------------------------------------------------------------------------

.ÿcontrastÿp.year,ÿpveffectsÿnowald

Contrastsÿofÿmarginalÿlinearÿpredictions

Marginsÿÿÿÿÿÿ:ÿasbalanced

-----------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿContrastÿÿÿStd.ÿErr.ÿÿÿÿÿÿzÿÿÿÿP>|z|
-------------+---------------------------------------
ÿÿÿÿÿÿÿÿyearÿ|
ÿÿÿ(linear)ÿÿ|ÿÿÿ.1054911ÿÿÿ.1493714ÿÿÿÿÿ0.71ÿÿÿ0.480
(quadratic)ÿÿ|ÿÿ-.0533967ÿÿÿ.1526225ÿÿÿÿ-0.35ÿÿÿ0.726
ÿÿÿÿ(cubic)ÿÿ|ÿÿ-.0130413ÿÿÿ.1520545ÿÿÿÿ-0.09ÿÿÿ0.932
ÿÿ(quartic)ÿÿ|ÿÿÿ-.006741ÿÿÿ.1499558ÿÿÿÿ-0.04ÿÿÿ0.964
ÿÿ(quintic)ÿÿ|ÿÿ-.0527641ÿÿÿ.1465866ÿÿÿÿ-0.36ÿÿÿ0.719
ÿÿÿ(sextic)ÿÿ|ÿÿÿ.0423435ÿÿÿ.1493729ÿÿÿÿÿ0.28ÿÿÿ0.777
ÿÿÿ(septic)ÿÿ|ÿÿ-.0078873ÿÿÿ.1520726ÿÿÿÿ-0.05ÿÿÿ0.959
ÿÿÿÿ(octic)ÿÿ|ÿÿ-.0385536ÿÿÿ.1518268ÿÿÿÿ-0.25ÿÿÿ0.800
ÿÿÿÿ(nonic)ÿÿ|ÿÿÿ.0409304ÿÿÿ.1545442ÿÿÿÿÿ0.26ÿÿÿ0.791
-----------------------------------------------------

.ÿ
.ÿ//ÿConfirmationÿthatÿtheÿregressionÿmodelÿfitsÿtheÿ"rawÿpercentages"
.ÿpredictÿdoubleÿxb,ÿxb

.ÿgraphÿtwowayÿscatterÿoutcomeÿyear,ÿmcolor(black)ÿmsize(small)ÿ||ÿ///
>ÿÿÿÿÿÿÿÿÿlineÿxbÿyear,ÿsortÿlcolor(black)ÿ///
>ÿÿÿÿÿÿÿÿÿylabel(ÿ,ÿangle(horizontal)ÿnogrid)ÿlegend(off)

.ÿ
.ÿ//ÿNoÿweighting,ÿonlyÿrankÿtransformation
.ÿnptrendÿoutcome,ÿby(year)

ÿÿÿÿÿÿÿÿÿÿyearÿÿÿÿÿÿÿÿscoreÿÿÿÿÿÿÿÿÿÿobsÿÿÿsumÿofÿranksÿÿ
ÿÿÿÿÿÿÿÿÿÿ1991ÿÿÿÿÿÿÿÿÿ1991ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ2ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1992ÿÿÿÿÿÿÿÿÿ1992ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1993ÿÿÿÿÿÿÿÿÿ1993ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ4ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1994ÿÿÿÿÿÿÿÿÿ1994ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿ10ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1995ÿÿÿÿÿÿÿÿÿ1995ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ3ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1996ÿÿÿÿÿÿÿÿÿ1996ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ6ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1997ÿÿÿÿÿÿÿÿÿ1997ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ7ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1998ÿÿÿÿÿÿÿÿÿ1998ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ8ÿÿ
ÿÿÿÿÿÿÿÿÿÿ1999ÿÿÿÿÿÿÿÿÿ1999ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ9ÿÿ
ÿÿÿÿÿÿÿÿÿÿ2000ÿÿÿÿÿÿÿÿÿ2000ÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ5ÿÿ

ÿÿÿÿÿÿÿÿÿÿzÿÿ=ÿÿ1.77
ÿÿProbÿ>ÿ|z|ÿ=ÿ0.076

.ÿ
.ÿexit

endÿofÿdo-file


.

Also, I'm not sure what you have against Stata commands (user-written ptrend or official nptrend) specifically written for tests of trend.

Thank you so much for your helpful reply! I actually don't have anything against the nptrend command—it works well—but the ptrend command seems to be weighting the observations based on the proportions and giving the "change" in the proportions rather than trends over time, am I right? or are both nptrend and ptrend accepted methods for trend analysis? I appreciate your response.