These are model solutions. Students will get full marks if they do at least the graphs suggested, reach the main conclusions, and justify their conclusions graphically.
If it takes more than one day of fasting to reduce plasma carotene to suitable base levels, baseline 2 readings will be lower than baseline 1. The scatterplot shows excellent agreement between the two, with points more or less evenly scattered above and below the diagonal, except for subject #71 (in row 1) whose baseline 1 was much higher than baseline 2. The histogram shows that the difference between the two baseline measures is tightly scattered about 0, except for one outlier (subject #71). Hence I would use either baseline 2 or an average of the two, but not baseline 1.
The following box plots of Week 12 minus baseline 1 and minus the mean baseline measure are almost the same. It is clear that Preparation 3 (BASF 30 mg) gives the greatest and most consistent elevation in plasma glucose levels. Preparation 4 (BASF 60 mg) is effective for some people, but overall is less consistent than Preparation 3. Preparation 2 (Roche 60 mg) is consistently the least effective.
There aren't many other graphs I could think of. This one shows the changes over 12 weeks relative to baseline 2 for each subject, with lines coded by preparation. It is not as simple to read as the boxplots. It shows that for most subjects, there was little change between weeks 6 and 12, suggesting that some treatments might achieve steady state levels of plasma carotene in less than 12 weeks.
The box plot of week 6 minus baseline 2 is similar to the boxplot for week 12 minus baseline 2, but the distribution for Preparation 3 is much more variable, suggesting that, for some subjects, Preparation 3 takes more than 6 weeks to achieve its final level.
> attach(betacar) > windows(h=4,w=4) > plot(bl1,bl2) > abline(0,1) > hist(bl1-bl2) > boxplot(split(wk12-bl2,prep),main="Week 12 - Baseline 2") > boxplot(split(wk12-(bl1+bl2)/2,prep),main="Week 12 - Mean Baseline") > matplot(seq(6,12,by=2),rbind(wk6-bl2,wk8-bl2,wk10-bl2,wk12-bl2),type="l",col=prep,lty=prep,xlab="Week",ylab="Plasma Carotene") > legend(7,350,paste("Prep",1:4),lty=1:4,col=1:4) > title("Change from Baseline 2") > boxplot(split(wk6-bl2,prep),main="Week 6 - Baseline 2")
The time series plots show annual cycles throughout and a decreasing trend beginning in 1990. This is reinforced by the boxplots by year. The boxplots by month suggest that highest concentrations of dieldrin in solids are found in the summer months.
The decrease since 1990 shows mostly in a decreasing minimum value, and since this is a very low concentration, the downward trend is much more evident on a log scale. The log scale also makes the variation within months much more comparable, that is, the boxplots by month are more uniform in hinge spread on the log scale than in original units.
The detection limits were 3.2 ng/g up to 1989-02-09 and 6.8 ng/g from 1989-05-25. It is clear from the time series plot that the 6.8 ng/g is an upper detection limit, but the 3.2 ng/g could be either an upper limit or a lower limit as there are both higher and lower values recorded in the same time period.
While the upper detection limits prevent us from seeing if the extreme high values are decreasing after 1989, it is still evident that the bottom of the annual cycle is getting lower.
Because extreme high values occur only a few times in most years, replacement of extreme high values by upper detection limits will not affect the annual medians and lower quartiles in the box plots by year, provided that the upper detection limits are much higher than the annual medians. This is the case from May 1989 onward, where the upper detection limit is 6.8 ng/g.
The Lag -1 plot shows moderate autocorrelation; on the log scale, the extreme high values have less impact and the scatter is more elliptical. The upper detection limits show as horizontal and vertical lines in the scatterplot.
> attach(diesol) > windows(h=4,w=6) > plot(julian,die.sol,type="l",xlab="Julian Day",ylab="Dieldrin in Solids") > plot(julian,die.sol,type="l",xlab="Julian Day",ylab="Dieldrin in Solids",log="y") > boxplot(split(die.sol,year),xlab="Year",ylab="Dieldrin in Solids") > boxplot(split(die.sol,year),xlab="Year",ylab="Dieldrin in Solids",log="y") > months <- c("Jan","Feb","Mar","Apr","May","Jun","Jul","Aug","Sep","Oct","Nov","Dec") > boxplot(split(die.sol,month)[months],ylab="Dieldrin in Solids") > boxplot(split(die.sol,month)[months],ylab="Dieldrin in Solids",log="y") > windows(h=4,w=4) > plot(die.sol[-length(die.sol)],die.sol[-1],xlab="Lag -1",ylab="Dieldrin in Solids",main="Lag -1 Plot") > abline(0,1) > plot(die.sol[-length(die.sol)],die.sol[-1],xlab="Lag -1",ylab="Dieldrin in Solids",main="Lag -1 Plot on Log Scale",log="xy") > abline(0,1)
> diesol[dl,] julian year month day die.sol dl 1 31504 1986 Apr 2 3.2 TRUE 2 31511 1986 Apr 9 3.2 TRUE 6 31539 1986 May 7 3.2 TRUE 11 31574 1986 Jun 11 3.2 TRUE 12 31588 1986 Jun 25 3.2 TRUE 15 31610 1986 Jul 17 3.2 TRUE 16 31617 1986 Jul 24 3.2 TRUE 17 31624 1986 Jul 31 3.2 TRUE 18 31631 1986 Aug 7 3.2 TRUE 19 31638 1986 Aug 14 3.2 TRUE 20 31645 1986 Aug 21 3.2 TRUE 21 31652 1986 Aug 28 3.2 TRUE 22 31659 1986 Sep 4 3.2 TRUE 23 31665 1986 Sep 10 3.2 TRUE 24 31673 1986 Sep 18 3.2 TRUE 27 31694 1986 Oct 9 3.2 TRUE 28 31701 1986 Oct 16 3.2 TRUE 29 31708 1986 Oct 23 3.2 TRUE 30 31715 1986 Oct 30 3.2 TRUE 31 31722 1986 Nov 6 3.2 TRUE 32 31730 1986 Nov 14 3.2 TRUE 33 31737 1986 Nov 21 3.2 TRUE 34 31743 1986 Nov 27 3.2 TRUE 35 31750 1986 Dec 4 3.2 TRUE 36 31757 1986 Dec 11 3.2 TRUE 37 31764 1986 Dec 18 3.2 TRUE 41 31806 1987 Jan 29 3.2 TRUE 42 31813 1987 Feb 5 3.2 TRUE 43 31820 1987 Feb 12 3.2 TRUE 44 31827 1987 Feb 19 3.2 TRUE 45 31834 1987 Feb 26 3.2 TRUE 46 31841 1987 Mar 5 3.2 TRUE 50 31870 1987 Apr 3 3.2 TRUE 59 31932 1987 Jun 4 3.2 TRUE 65 31981 1987 Jul 23 3.2 TRUE 67 31995 1987 Aug 6 3.2 TRUE 69 32009 1987 Aug 20 3.2 TRUE 79 32079 1987 Oct 29 3.2 TRUE 89 32191 1988 Feb 18 3.2 TRUE 99 32297 1988 Jun 3 3.2 TRUE 103 32324 1988 Jun 30 3.2 TRUE 105 32366 1988 Aug 11 3.2 TRUE 106 32373 1988 Aug 18 3.2 TRUE 123 32499 1988 Dec 22 3.2 TRUE 124 32507 1988 Dec 30 3.2 TRUE 126 32520 1989 Jan 12 3.2 TRUE 127 32527 1989 Jan 19 3.2 TRUE 129 32541 1989 Feb 2 3.2 TRUE 130 32548 1989 Feb 9 3.2 TRUE 141 32653 1989 May 25 6.8 TRUE 169 32856 1989 Dec 14 6.8 TRUE 173 32898 1990 Jan 25 6.8 TRUE 188 33010 1990 May 17 6.8 TRUE 201 33101 1990 Aug 16 6.8 TRUE 202 33108 1990 Aug 23 6.8 TRUE 203 33115 1990 Aug 30 6.8 TRUE 212 33185 1990 Nov 8 6.8 TRUE 244 33437 1991 Jul 18 6.8 TRUE 245 33444 1991 Jul 25 6.8 TRUE 246 33451 1991 Aug 1 6.8 TRUE 247 33458 1991 Aug 8 6.8 TRUE 249 33472 1991 Aug 22 6.8 TRUE 250 33479 1991 Aug 29 6.8 TRUE 251 33486 1991 Sep 5 6.8 TRUE 253 33507 1991 Sep 26 6.8 TRUE 254 33514 1991 Oct 3 6.8 TRUE 255 33521 1991 Oct 10 6.8 TRUE 256 33528 1991 Oct 17 6.8 TRUE 258 33542 1991 Oct 31 6.8 TRUE 262 33570 1991 Nov 28 6.8 TRUE 290 33773 1992 Jun 18 6.8 TRUE 291 33780 1992 Jun 25 6.8 TRUE 293 33794 1992 Jul 9 6.8 TRUE 294 33801 1992 Jul 16 6.8 TRUE 295 33808 1992 Jul 23 6.8 TRUE 297 33815 1992 Jul 30 6.8 TRUE 299 33829 1992 Aug 13 6.8 TRUE 301 33836 1992 Aug 20 6.8 TRUE 302 33843 1992 Aug 27 6.8 TRUE 303 33850 1992 Sep 3 6.8 TRUE 304 33857 1992 Sep 10 6.8 TRUE 305 33864 1992 Sep 17 6.8 TRUE 306 33871 1992 Sep 24 6.8 TRUE 307 33878 1992 Oct 1 6.8 TRUE 331 34060 1993 Apr 1 6.8 TRUE 333 34074 1993 Apr 15 6.8 TRUE 334 34081 1993 Apr 22 6.8 TRUE 342 34137 1993 Jun 17 6.8 TRUE 345 34158 1993 Jul 8 6.8 TRUE 353 34214 1993 Sep 2 6.8 TRUE 357 34242 1993 Sep 30 6.8 TRUE 365 34298 1993 Nov 25 6.8 TRUE 391 34509 1994 Jun 24 6.8 TRUE 402 34592 1994 Sep 15 6.8 TRUE 430 34830 1995 May 11 6.8 TRUE 445 34934 1995 Aug 23 6.8 TRUE 451 34977 1995 Oct 5 6.8 TRUE 452 34984 1995 Oct 12 6.8 TRUE 475 35152 1996 Mar 28 6.8 TRUE