Glover, Tim

2017-Sep-05 13:24 UTC

### [R] Interesting behavior of lm() with small, problematic data sets

I've recently come across the following results reported from the lm() function when applied to a particular type of admittedly difficult data. When working with small data sets (for instance 3 points) with the same response for different predicting variable, the resulting slope estimate is a reasonable approximation of the expected 0.0, but the p-value of that slope estimate is a surprising value. A reproducible example is included below, along with the output of the summary of results ######### example code x <- c(1,2,3) y <- c(1,1,1) #above results in{ (1,1) (2,1) (3,1)} data set to regress new.rez <- lm (y ~ x) # regress constant y on changing x) summary(new.rez) # display results of regression ######## end of example code Results: Call: lm(formula = y ~ x) Residuals: 1 2 3 5.906e-17 -1.181e-16 5.906e-17 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.000e+00 2.210e-16 4.525e+15 <2e-16 *** x -1.772e-16 1.023e-16 -1.732e+00 0.333 --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 Residual standard error: 1.447e-16 on 1 degrees of freedom Multiple R-squared: 0.7794, Adjusted R-squared: 0.5589 F-statistic: 3.534 on 1 and 1 DF, p-value: 0.3112 Warning message: In summary.lm(new.rez) : essentially perfect fit: summary may be unreliable ############## There is a warning that the summary may be unreliable sue to the essentially perfect fit, but a p-value of 0.3112 doesn?t seem reasonable. As a side note, the various r^2 values seem odd too. Tim Glover Senior Scientist II (Geochemistry, Statistics), Americas - Environment & Infrastructure, Amec Foster Wheeler 271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105 T +01 978 692 9090 D +01 978 392 5383 M +01 850 445 5039 tim.glover at amecfw.com amecfw.com This message is the property of Amec Foster Wheeler plc and/or its subsidiaries and/or affiliates and is intended only for the named recipient(s). Its contents (including any attachments) may be confidential, legally privileged or otherwise protected from disclosure by law. Unauthorised use, copying, distribution or disclosure of any of it may be unlawful and is strictly prohibited. We assume no responsibility to persons other than the intended named recipient(s) and do not accept liability for any errors or omissions which are a result of email transmission. If you have received this message in error, please notify us immediately by reply email to the sender and confirm that the original message and any attachments and copies have been destroyed and deleted from your system. If you do not wish to receive future unsolicited commercial electronic messages from us, please forward this email to: unsubscribe at amecfw.com and include ?Unsubscribe? in the subject line. If applicable, you will continue to receive invoices, project communications and similar factual, non-commercial electronic communications. Please click http://amecfw.com/email-disclaimer for notices and company information in relation to emails originating in the UK, Italy or France.

Jeff Newmiller

2017-Sep-05 16:05 UTC

### [R] Interesting behavior of lm() with small, problematic data sets

Why does an unreliable fit have to provide "reasonable" results? More specifically, p-values arise from observed distributions... if your slopes are "in the noise" then the slope estimate's location within that distribution could be anywhere relative to the center and spread of that very narrow distribution, leading to, ah, what was it... oh, right... "unreliable" results. -- Sent from my phone. Please excuse my brevity. On September 5, 2017 6:24:30 AM PDT, "Glover, Tim" <Tim.Glover at amecfw.com> wrote:>I've recently come across the following results reported from the lm() >function when applied to a particular type of admittedly difficult >data. When working with >small data sets (for instance 3 points) with the same response for >different predicting variable, the resulting slope estimate is a >reasonable approximation of the expected 0.0, but the p-value of that >slope estimate is a surprising value. A reproducible example is >included below, along with the output of the summary of results > >######### example code >x <- c(1,2,3) >y <- c(1,1,1) > >#above results in{ (1,1) (2,1) (3,1)} data set to regress > >new.rez <- lm (y ~ x) # regress constant y on changing x) >summary(new.rez) # display results of regression > >######## end of example code > >Results: > >Call: >lm(formula = y ~ x) > >Residuals: > 1 2 3 > 5.906e-17 -1.181e-16 5.906e-17 > >Coefficients: > Estimate Std. Error t value Pr(>|t|) >(Intercept) 1.000e+00 2.210e-16 4.525e+15 <2e-16 *** >x -1.772e-16 1.023e-16 -1.732e+00 0.333 >--- >Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 > >Residual standard error: 1.447e-16 on 1 degrees of freedom >Multiple R-squared: 0.7794, Adjusted R-squared: 0.5589 >F-statistic: 3.534 on 1 and 1 DF, p-value: 0.3112 > >Warning message: >In summary.lm(new.rez) : essentially perfect fit: summary may be >unreliable > > >############## > >There is a warning that the summary may be unreliable sue to the >essentially perfect fit, but a p-value of 0.3112 doesn?t seem >reasonable. >As a side note, the various r^2 values seem odd too. > > > > > > > >Tim Glover >Senior Scientist II (Geochemistry, Statistics), Americas - Environment >& Infrastructure, Amec Foster Wheeler >271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105 >T +01 978 692 9090 D +01 978 392 5383 M +01 850 445 5039 >tim.glover at amecfw.com amecfw.com > > >This message is the property of Amec Foster Wheeler plc and/or its >subsidiaries and/or affiliates and is intended only for the named >recipient(s). Its contents (including any attachments) may be >confidential, legally privileged or otherwise protected from disclosure >by law. Unauthorised use, copying, distribution or disclosure of any of >it may be unlawful and is strictly prohibited. We assume no >responsibility to persons other than the intended named recipient(s) >and do not accept liability for any errors or omissions which are a >result of email transmission. If you have received this message in >error, please notify us immediately by reply email to the sender and >confirm that the original message and any attachments and copies have >been destroyed and deleted from your system. If you do not wish to >receive future unsolicited commercial electronic messages from us, >please forward this email to: unsubscribe at amecfw.com and include >?Unsubscribe? in the subject line. If applicable, you will continue to >receive invoices, project communications and similar factual, >non-commercial electronic communications. > >Please click http://amecfw.com/email-disclaimer for notices and company >information in relation to emails originating in the UK, Italy or >France. > >______________________________________________ >R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see >https://stat.ethz.ch/mailman/listinfo/r-help >PLEASE do read the posting guide >http://www.R-project.org/posting-guide.html >and provide commented, minimal, self-contained, reproducible code.

David Winsemius

2017-Sep-05 16:28 UTC

### [R] Interesting behavior of lm() with small, problematic data sets

> On Sep 5, 2017, at 6:24 AM, Glover, Tim <Tim.Glover at amecfw.com> wrote: > > I've recently come across the following results reported from the lm() function when applied to a particular type of admittedly difficult data. When working with > small data sets (for instance 3 points) with the same response for different predicting variable, the resulting slope estimate is a reasonable approximation of the expected 0.0, but the p-value of that slope estimate is a surprising value. A reproducible example is included below, along with the output of the summary of results > > ######### example code > x <- c(1,2,3) > y <- c(1,1,1) > > #above results in{ (1,1) (2,1) (3,1)} data set to regress > > new.rez <- lm (y ~ x) # regress constant y on changing x) > summary(new.rez) # display results of regression > > ######## end of example code > > Results: > > Call: > lm(formula = y ~ x) > > Residuals: > 1 2 3 > 5.906e-17 -1.181e-16 5.906e-17 > > Coefficients: > Estimate Std. Error t value Pr(>|t|) > (Intercept) 1.000e+00 2.210e-16 4.525e+15 <2e-16 *** > x -1.772e-16 1.023e-16 -1.732e+00 0.333 > --- > Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 > > Residual standard error: 1.447e-16 on 1 degrees of freedom > Multiple R-squared: 0.7794, Adjusted R-squared: 0.5589 > F-statistic: 3.534 on 1 and 1 DF, p-value: 0.3112 > > Warning message: > In summary.lm(new.rez) : essentially perfect fit: summary may be unreliable > > > ############## > > There is a warning that the summary may be unreliable sue to the essentially perfect fit, but a p-value of 0.3112 doesn?t seem reasonable. > As a side note, the various r^2 values seem odd too.You have an overfitted model with only 3 perfectly fit-able data points and you are whinging about a Wald statistic about which you were warned. I think you are wasting our time. (But I'm fully retired and I have a lot of time to waste.) I seem to remember that a t-distribution with 1 degree of freedom is actually the Cauchy distribution. I would point out that you can also get:> 2*pt(-1.732e+00, 1)[1] 0.3333414 So maybe from that perspective any value might be "reasonable" from the perspective that you have that particular number data points (so one degree of freedom) and are using an estimate of the t-statistic which is essentially the ratio of 0/0 from a numerical point of view. -- David.

mark.hogue at srs.gov

2017-Sep-05 16:31 UTC

### [R] Interesting behavior of lm() with small, problematic data sets

Tim, I think what you're seeing is https://en.wikipedia.org/wiki/Loss_of_significance. Cheers, Mark From: "Glover, Tim" <Tim.Glover at amecfw.com> To: "r-help at r-project.org" <r-help at r-project.org> Date: 09/05/2017 11:37 AM Subject: [R] Interesting behavior of lm() with small, problematic data sets Sent by: "R-help" <r-help-bounces at r-project.org> I've recently come across the following results reported from the lm() function when applied to a particular type of admittedly difficult data. When working with small data sets (for instance 3 points) with the same response for different predicting variable, the resulting slope estimate is a reasonable approximation of the expected 0.0, but the p-value of that slope estimate is a surprising value. A reproducible example is included below, along with the output of the summary of results ######### example code x <- c(1,2,3) y <- c(1,1,1) #above results in{ (1,1) (2,1) (3,1)} data set to regress new.rez <- lm (y ~ x) # regress constant y on changing x) summary(new.rez) # display results of regression ######## end of example code Results: Call: lm(formula = y ~ x) Residuals: 1 2 3 5.906e-17 -1.181e-16 5.906e-17 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.000e+00 2.210e-16 4.525e+15 <2e-16 *** x -1.772e-16 1.023e-16 -1.732e+00 0.333 --- Signif. codes: 0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1 Residual standard error: 1.447e-16 on 1 degrees of freedom Multiple R-squared: 0.7794, Adjusted R-squared: 0.5589 F-statistic: 3.534 on 1 and 1 DF, p-value: 0.3112 Warning message: In summary.lm(new.rez) : essentially perfect fit: summary may be unreliable ############## There is a warning that the summary may be unreliable sue to the essentially perfect fit, but a p-value of 0.3112 doesn?t seem reasonable. As a side note, the various r^2 values seem odd too. Tim Glover Senior Scientist II (Geochemistry, Statistics), Americas - Environment & Infrastructure, Amec Foster Wheeler 271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105 T +01 978 692 9090 D +01 978 392 5383 M +01 850 445 5039 tim.glover at amecfw.com amecfw.com This message is the property of Amec Foster Wheeler plc and/or its subsidiaries and/or affiliates and is intended only for the named recipient(s). Its contents (including any attachments) may be confidential, legally privileged or otherwise protected from disclosure by law. Unauthorised use, copying, distribution or disclosure of any of it may be unlawful and is strictly prohibited. We assume no responsibility to persons other than the intended named recipient(s) and do not accept liability for any errors or omissions which are a result of email transmission. If you have received this message in error, please notify us immediately by reply email to the sender and confirm that the original message and any attachments and copies have been destroyed and deleted from your system. If you do not wish to receive future unsolicited commercial electronic messages from us, please forward this email to: unsubscribe at amecfw.com and include ?Unsubscribe? in the subject line. If applicable, you will continue to receive invoices, project communications and similar factual, non-commercial electronic communications. Please click http://amecfw.com/email-disclaimer for notices and company information in relation to emails originating in the UK, Italy or France. ______________________________________________ R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. [[alternative HTML version deleted]]

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