Genomics

I bet you know the words to Three Blind Mice and Hickory Dickory Doc and have seen at least some of Mighty Mouse, Tom & Jerry, Speedy Gonzales, An American Tail, Stuart Little, The Great Mouse Detective, Flushed Away, Ratatouille, and Desperaux. You’ve hugged Mickey and Minnie at Disney World. You’ve played the ultimate Rube Goldberg game Mouse Trap (and maybe even used the very handy 2D graphical JSL function with the same name). You have a fancy electronic one attached to your computer. You’ve read (Cliffs or Spark notes for) John Steinbeck’s classic novel. A neutral observer of Western culture over the last century might easily conclude we love mice.

But hang on. Over the centuries, these furry little friends have shuttled countless fleas bearing Yersinia pestis, the microbe of bubonic plague (the Black Death), killing around one-third of Europe between 1347 to 1353 and causing numerous devastating outbreaks ever since.

On a less severe but likely equally numerous scale, they’ve exasperated wives, husbands and children in homes everywhere. A $&*# one even found its way into our pantry last week; that’s not supposed to happen in 2009! Key science question: Why does it nibble daintily in bags of powdered sugar, chocolate candies and caramel but doesn’t touch the Fiber One or wheat bran? Regardless, an old-school trap baited with peanut butter summarily ended its mini-foray into food paradise.

What about the mouse’s bigger and tougher cousin, the rat? Who really loves or wants to be a rat? James Cagney, Willard, and Ratigan certainly don’t help. My father-in-law, who spent much of his career in a naval shipyard, tells stories of rats the size of felines and canines, and meaner than both.

So do we love or hate mice?

Genomics will tip the scales in their favor. The little critters offer great promise to human health as a model organism, as the mouse genome is around 99% similar to ours. Commercial, government and academic institutions worldwide are making great strides in mouse comparative genomics; see the National Human Genome Research Institute Web site for some details. Top-tier researchers like David Threadgill (formerly at UNC, now at NC State) have provided great service to the scientific community with their extensive inbred lines and genetic analyses. The links between mice and men are often difficult and error prone, but steady progress continues (Arriba, Arriba, Andele! Andele!).

At SAS, we’re hoping to help with new features in JMP Genomics 4.1 (due out late this year) that provide graphical methods for interactively viewing and comparing advanced statistical results across an entire mouse or rat genome, including ability to drill down and display tracks.

Like Reepicheep in The Chronicles of Narnia, a hero of the 21st century could very well be one with an already-cool name: Mus musculus.

We are privileged to live in an age of advanced science and technology. Software is one of the most amazing parts of it, providing complex programs to control mind-boggingly incredible inventions that help us with such things as unraveling the deep mysteries of genomics. As widesweeping as software has become, it’s important to keep it in perspective with respect to our interpersonal relationships and worldview. A fun trip I had this past weekend helped remind me of this.

The vessel: A 36-foot Catalina Sloop Rig

The voyagers:

• Pops, wise skipper and veterinarian
• Daniel, thoughtful son of Pops, Penn State devotee, and trip organizer
• Eric, fluent in Greek, Latin, and Phillies baseball
• Ben, DC native with an encyclopedic mind for trivia
• Yours truly

The voyage: From Bowley’s Quarters, Maryland (20 nautical miles north of Baltimore), to the Inner Harbor and a Saturday evening Orioles game at Camden Yards, fulfilling one of Ben’s lifelong dreams.

In critical need of stout nourishment before embarking on the Chesapeake, we frequented a converted double-wide trailer in the sticks a few miles inland. Plush accommodations consisted of lawn chairs and folding tables covered with plastic. In a few minutes, our waitress (who appeared to have been on a few voyages of her own) plopped down husks of fresh sweet corn, a pitcher of light beer and a pot of bright red steamed crabs, claws intertwined. Under expert guidance from Ben, we systematically dismantled the crustaceans and extracted the tasty edibles. With plenty of laughs and wounded fingers smarting with Old Bay seasoning, it was the perfect start.

We hoisted our pirate flag and set sail at dawn (Pacific time). Although routine by today’s standards, Pops’ sailboat was impressive and undoubtedly would have made brown sailors of old green with envy. The ropes were all neatly pulleyed; the main sail was raised easily with one of them and the ferruled jib simply unrolled with another. The galley came complete with refrigerator and freezer (ice cream!), stove, oven and sink. Every nook and cranny was purposefully utilized. Our electronic gear instantly informed of us depth of water, boat speed, exact GPS coordinates, and wind speed and direction. We were really roughing it.

It was a glorious sunny day in the 80s with a stiff breeze against us. Via the magic of Bernoulli forces on our two sails and Pops’ skillful navigation, we steadily tacked our way southward. To trim the jib on each side, we alternately cranked two nicely engineered double ratcheted winches. The job was easy enough that Ben and I dozed sometimes in between tacks, earning ourselves the nickname “the napping wenches.”

My primary duty was to spot crab-pot buoys, bestrung across the bay like large waterbugs. We only ran over two of them, which statistically I thought was excellent work, but the crew threatened me with a keelhauling if we hit any more. This is the practice of tying an unfortunate soul about the waist with ropes and dragging them under and across the barnacle-laden hull of the ship, iterating as necessary.

I was also responsible for lunch that day. Being in possession of the adverse allele of the CULN2 gene, I had managed the day before to bribe two of my daughters to do the honors. As Food Channel junkies, they came through in grand style with elegant ham wraps and a gourmet pasta salad. Catastrophe struck as a few of the wraps found their way to ice water in the cooler, but with the aid of one of Pops’ handy fans they were saved -- and shipmates none the wiser.

After several hours, we conquered the choppy water and crossed triumphantly under the Francis Scott Key Bridge. As if on cue, a very loud report from one of the cannons at Fort McHenry signaled our arrival and nearly startled Ben overboard. We passed several massive cargo ships, one with the sweet smell of brown sugar being scooped in huge buckets onto the conveyor chute of the Domino’s Sugar factory. We arrived with time to saunter through the chic Baltimore Inner Harbor condos and attractions and to wind our way up to Camden Yards.

At the ballpark during batting practice, there is keen competition to grab balls hit over the fences. With strategic anticipation sharpened on the high seas, Eric and Daniel managed to snag three. The game itself was rousing and close, but the Orioles lost their lead near the end. Ben felt a curse and said he hadn’t been present for a win in this stadium for seven years. Then in the bottom of the 12th, Mora cracked a spectacular walk-off homer, and the dream was realized. Eric, watching real-time cell-phone updates, was more pumped that the Phillies came back from a 5-0 deficit to win 8-7 in the ninth. Even my poor Cleveland Indians eked out a victory, and a night of baseball perfection was complete. We celebrated back on the dock, staying up way past bedtime.

On the return trip the next day, alas, the winds were sucked to a standstill by Scylla and Kharybdis. We reluctantly overcame them with the iron mainsail (compliments of Rudolf Diesel), and even had time to anchor offshore for awhile and cannonball off of the back of the boat.

This trip was a great mix of old and new, of nature and gadgetry. The best part was the camaraderie and conversation. We had some treasurable time for philosophy and theology, peppered with Nittany Lion – Buckeye trash talk. (Those with Y chromosomes usually require such things before they can open up.) Technology rocks, but as sailors of all time will attest, there’s no substitute for seamates.

This weekend features one of my all-time favorite sporting events: the golf US Open (plus Father’s Day on Sunday provides a convenient guilt-free excuse to actually watch it). This year, the tournament is held at the Bethpage Black course just outside of New York City. It has a classic sign:

Not sure who “We” represents, but there is an unquestionable tone of authority, with prepositions, definite and indefinite articles all capitalized. And note this course is not just for “Skilled Golfers”, but “Highly Skilled Golfers.” (Hats off to the pros this week who are also battling Mother Nature under very soggy conditions.)

Some data sets should come with a similar warning. In genomics, we are now faced with experiments conducted on thousands of individuals with millions of measurements on each across a variety of complex molecular domains: genetic markers, transcript abundance, copy number, microRNA, protein and metabolite intensities, not to mention thousands of standard phenotypes. Analyzing such data sets properly certainly requires skill along with the best possible software.

A primary goal of JMP since its inception more than 20 years ago has been to provide a dynamic and optimal combination of both statistics and graphics. Drop one of this pair, and you are going to miss something critical. JMP Genomics, although much younger, is definitely building on the same philosophy. Accomplishing this goal is difficult, but we continue to make progress and relish your feedback on how to do it better.

Confession: Every time I pass a mirror and find no one is looking, I practice my golf swing. I’m a sucker for every newfangled idea on how to hit a golf ball better -- trust me, there is an infinite supply of them -- and just have to try it. I’ve been tinkering with my dang swing since graduate school and still don’t have it right. Not seeing him enter, I almost knocked a guy out one time with my mock follow-through in a small men’s room.

One technology that’s a godsend is digital video with slow motion. Interactively successive freeze frames taken from good angles show exactly what I’m doing (even though I think I’m doing something else) -- that, of course, and actually striking that dumb little 1.68 inch diameter sphere and adding up my score.

Pictures and Numbers. One day they might even get me to “Highly Skilled.”

For about a year now, I’ve been having trouble with my faithful Craftsman Eager-1 push mower, which I bought at Sears about 15 years ago. The daggum thing starts up fine but then cuts off after about 2 seconds. This has led me to adopt the following algorithm:

1. Press rubber priming balloon 5 times.
2. Pull rip cord to start engine.
3. If the engine does not start, go to Step 1.
4. If the engine starts but then stalls after a few seconds, utter expletives go to Step 1.
5. If the engine stays on, mow the grass.
6. If the engine cuts off again, utter stronger expletives and go back to Step 1.

My problem is that the number of iterations through Steps 1-4 has been steadily increasing by about 2 times per month. Now that I’m north of 20 times for just one mow, our longstanding relationship is really on the rocks. I think it’s time for a new mower, but what am I to do with this one?

On my way to the store to check out the latest new models (urggh-urggh), I pass a small house with a hand-written sign out front:

Good Used Lawn Mowers
555-1234

“What the heck,” I say to myself as I pop the number into my cell. After a few rings, an elderly voice with a friendly Southern drawl answers:

“Hello.”

“Hi, I’ve been having trouble with my mower and was wondering if you might be interested in it.”

“Bring it by on Saturday at 9 in the mornin’, ‘cause I like to sleep in on weekends. When you come, knock on the back door.”

“OK, see you then.”

On Saturday, I pull around behind the house to find a double-wide carport with what must be the largest assemblage of used hand mowers in the state of North Carolina. Some are obviously very old but are neatly arranged, and all appear to be ready for action.

I knock on the door, and a spry old gentlemen with a flannel shirt and workman pants pulled up over his stomach with suspenders greets me with a firm handshake and beckons me to an overstuffed chair in his small living room. We exchange pleasantries, and I learn he is 91 years old and had spent 45 years as a railroad engineer. He has lived in Cary his entire life, and after his wife passed away, he started tinkering with mowers.

We go out back to take a look at the Eager-1. He deftly removes the air filter and squirts a bit of starting fluid into the exposed hole.

“Give ’r a pull,” he tells me.

I obey, and the Tecumseh engine roars to life but then quickly begins to stall in its usual fashion. But just before it completely dies, he gently places his index finger over the hole and, lo-and-behold, the engine coughs and cycles back to full power! It tries to stall again, but with perfect timing he chokes off the hole just enough to maintain that magic mixture of fuel and air. After a few more taps the engine is running steadily and better than ever. “Give ’r a try,” he says, pointing to a patch of grass. I am so ecstatic that I mow his whole backyard.

As software users and developers, we’re often tempted to abandon and bash old technologies and go for the latest, greatest new thing. While I’m certainly in favor of using the best means possible for the task at hand, sometimes those best means are those that have stood the test of time and have benefitted from the wisdom of those who have struggled through and solved myriads of problems using them. I put classic SAS software into this category – it provides a richly deep and powerful foundation for the processes in JMP Genomics. Our team continues to learn about clever new ways to use it to more effectively handle genomics data.

By the way, using this new-to-me-but-really-old-school technique, I can now start my Eager-1 with a single pull . It’s still going strong, and so is my nonagenarian mower buddy.

Thanks for the enlightening comments to my blog post "I Like 3-D Pie Charts" and for the new graphs. While the bar charts from Joe and John are very nice, I prefer vertical bars because of their connection with the gravity orientation of trees, mountains, buildings and of course cell-phone signal strength. It’s interesting that Paige and Lee suggest 2D pie charts. Warning: The graphics gods are watching you. Daniel, I’m glad you did not sleep through Art Appreciation in college.

SAS is well known for its corporate amenities, and its cafés are no exception. I was in line for lunch the other day and came upon the always tempting dessert case. The middle shelf featured over a dozen haphazardly arranged pieces of another one of my all-time favorites: chocolate crème pie. Simply irresistible. Maybe it’s my upbringing, but I had absolutely no problem identifying the largest slice. I don’t think it’s a guy thing either, because in the extra second I took reveling about how good it was going to look on my tray, the woman behind me handily placed it on hers. How dare she! But I was smiling inwardly as I quickly grabbed the next largest one, not only because we were likely identical by state for a latent canine quantitative trait, but because I had some more assurance that at least some humans are actually decent at assessing the size of three-dimensional wedge shapes. The psychological experimental evidence cited to the contrary is largely conducted on college students who are really only good at determining how much liquid remains in bottle-shaped objects.

This has me wondering if exploding pieces of the pie chart might actually be helpful in avoiding volumetric distortion. If nothing else, exploding and appropriately labeling one or more pieces seems like a great way to emphasize them, and, conversely, leaving the really thin slices unlabeled is desirable when they are not of interest.

Regarding ordering of slices or bars, I forgot that in examples like this one there is an overriding analytical criterion. In statistical modeling in general, one typically puts main effects first, then two-way interactions, then succeeding higher order interactions and finally residual/unexplained variance. One way to accomplish this (or any other ordering) in JMP is with the Value Ordering column property. For our example, if you first run JSL code like the following…

vc_col = Column("Variance Component");

vc_col << Set Property(Value Ordering, {"Array", "Line", "Sex", "Age", 


"Line*Sex", "Line*Age", "Sex*Age", "Line*Sex*Age", "Residual"});

…then the bars appear in the desired order. We use such a column property in JMP Genomics scripts to make sure chromosome plots appear in numerical instead of alphabetical order.

If you’re having trouble comparing sizes of slices in the next pie chart you see, just pretend that it’s your favorite dessert and you’re really hungry. Works every time.

So you know I’m a faithful left-brained statistician who makes every attempt to adhere to the highest professional standards of data visualization and analysis. Graphics luminaries like Edward Tufte and Stephen Few have made very valuable contributions to the field, and I bow to their wisdom.

But I have a secret confession to make: I like three-dimensional pie charts. It’s wrong, and I don’t know why; I just like them. Actually, I may be starting to figure out why. (BTW, new research reveals that a telltale sign of having at least one geek allele is a preponderance to begin sentences with the word “actually”—we’re searching for the corresponding genes with JMP Genomics.)

We know the commandments from the graphics gods: Keep it simple. Avoid chart junk. Let the data shine through. Favor linear over spatial comparisons. Eschew volumetric distortion. Wield Ockham’s Razor. Obey these commandments at all times. 3-D pie charts are the worst offenders and have long ago been banished to graph purgatory. Few has explained.

Why do I still like them? Let me be your graph optometrist for a sec and ask that annoyingly simple question: “Better A or Better B?”

So ... Better A?



Or ... Better B?



While a usual reply to my optometrist is “I can’t freakin' tell because of those stinging drops you just put in my eyes,” in this case, the answer for me is B.

Some more background: The purpose here is to quickly and effectively convey the dominant sources of variation in a microarray experiment. Without doubt, the bar chart A has more detail and nicely uses linear instead of spatial comparison. It’s a great graph and in fact is the default one shown for such analyses in JMP Genomics.

Why B? It takes advantage of color, aggregation and 3-D aesthetics. The labels enable immediate identification with the data instead of forcing me to eyeball down to an X-axis and tilt the head to read them. In addition, blogosphere exigencies require omission of a critical feature: interactivity. It’s a spinnable graph that comes complete with slider bars that let you adjust degree of explosion and shininess. (Thanks to JMP experts Xan Gregg, who has written about 3-D pie charts in JMP, Craige Hales and David Barbour.) The ability to personally control the graph won me over. Graph B also appears to be better suited for rapid scan viewing as recommended by Bill Cleveland. Heaven forbid: multiple 3-D pie charts!

Could it be my artistic right brain has suddenly come to life like a vampire after decades of dormancy and is in need of a consultation with Van Helsing?

This all has something to do with philosophical presuppositions. Dutch philosopher Herman Dooyeweerd and colleagues have extensively discussed 14 Aspects of Reality arranged in a specific order:

1. Numerical
2. Spatial
3. Kinematic
4. Physical
5. Biotical
6. Sensitive-psychical
7. Logical
8. Cultural-historical
9. Social
10. Economical
11. Aesthetical
12. Juridicial
13. Ethical
14. Fiducial

They make a convincing case that these ordered aspects are irreducible in the sense that you cannot eliminate any of them without getting into irrecoverable binds and self-refuting contradictions. Furthermore, nearly all philosophical conflicts throughout history have arisen from different attempts to make one of these aspects the divine/ultimate one upon which all others depend. (Such reductions have often turned Ockham’s Razor into Sweeney Todd’s.) Although there is a lot more to it, everything in creation possesses each of these aspects in varying degrees. For example, the computer on which you are reading this blog exists in space, has physical properties, has economic value, etc.

With reference to the bar and pie charts above, the bar chart relies primarily on the numerical, spatial and logical aspects, whereas the interactive pie chart adds aesthetics and kinematics. These latter two aspects make a big difference and enable the pie chart to connect with the viewer on more levels. We’re naturally drawn to things that are beautiful and exhibit pleasing colors, symmetry,and interactivity. We travel the world to engage with captivating wonders and works of art, both natural and man-made. We reward business professionals and politicians who build their careers not on the substance of their message, but by the elegance and flair with which they convey it. We play Guitar Hero and Rock Band for hours on end.

The pie chart also offers a biotic connection to various round delectables. More confessions: My wife makes the world’s best grated-apple pie, and I grew up devouring my mom’s to-die-for strawberry pie. I love pizza and cheesecake and even eat quiche from time to time. So I’m environmentally conditioned to be sorely tempted by the evil 3-D pie chart, and I’ve succumbed.

So you still prefer a bar chart? That’s fine; the gods are pleased. For now, I’m cranking up Warrant’s “Cherry Pie” on my iPod and playing with some more interactive 3-D graphics.

My youngest daughter was recently watching a rerun of Home Improvement featuring Tim “The Tool Man” Taylor and his mock TV show Tool Time. She said “Dad, you and Tim both seem to get into a lot of trouble when playing with power tools.” My wife added that throwing a statistical degree into the mix appears to have increased the trouble coefficient. My vehement objections were met with a reminder of several recent incidents, including how we spent most of New Year’s Eve this year without water due to a slight mishap with copper joint I had sweated in a downstairs bathroom renovation project. OK, so I’m a DIY junkie and have a garage full of tools, but the operative word in this instance is slight.

Of course no home project would get very far without a power drill, and I have three: an 18V cordless with dual batteries, an old-school 3/8” Craftsman that works great whenever those accursed batteries weaken, and a red 1/2” Milwaukee complete with side handle and heavy duty chuck. The Milwaukee is great for man stuff like drilling holes in railroad ties and mixing big batches of thinset mortar. Tim would give it an “Urghh Urghh” (my feeble attempt to spell his signature simian/caveman/Scooby-Do/Chewbacca grunt) .

Which brings us to the topic of this blog: Drill Down. Genomics data sets typically require conducting tens of thousands to millions (and in some cases even billions) of statistical tests, and we are immediately faced with how to most effectively present and explore the most significant ones. The idea behind Drill Down is to start with a summary graphic that highlights interesting points and enables you interactively zoom on them and obtain more statistical or graphical detail hierarchically.

A prototypical example of such a summary graph is a volcano plot, e.g. as generated by JMP Genomics’ One-Way ANOVA Analytical Process:

Each point represents a gene. The X-axis charts some measure of change (e.g. a difference between least-squares means) and the Y-axis is the corresponding –log10 p-value. The plot has a characteristic V-shape because of the statistical relationship between X and Y. The most significant points are towards the top and resemble lava rocks erupting from a volcano [insert Tim Taylor grunt here] and the least significant ones--a vast majority--are nicely overstruck at the bottom. The plots are dynamically linked to other displays like dendrograms and parallel plots, complete with brushing and zooming, capabilities not available in R. (Historical note: I believe Greg Gibson and I coined the term “volcano plot” in a 2001 Nature Genetics paper.)

After selecting desired points in the volcano, go to the window of “Action Buttons”, which offers a variety of Drill Down capabilities:

Drill Down with such Action Buttons is different from zoom a la Google Earth; you can launch sophisticated statistical analyses on the selected points, taking full advantage of either SAS or JMP functionality. For example, “Fit Model to Input Data for Selected Rows and Plot LS Means” launches JMP’s Fit Model platform on the selected genes (using the same model that you originally specified) and produces an informative integrated mix of statistical tables and graphics. Clicking “Plot OneWay Means by Chromosome and Position” fits one-way models in a SAS Data Step and then generates graphs like the following:

In this case the expression profiles of three treatment conditions are plotted along a very short section of the genome. The width of the confidence bands is interactively adjustable and the plot is embellished with gene track annotation that was automatically incorporated from a supplemental file specified in the initial JMPG dialog. You can drill down further and bring up various web pages corresponding to genes of interest.

So “Urghh Urghh” for Drill Down! It’s available in a few other places in JMP Genomics (e.g. Multidimensional Scaling and Predictive Modeling) and you can expect more in future releases.

By the way, that bathroom project is now finished and turned out beautifully. All three of the drills came in handy (if you want to borrow the Milwaukee please let me know). It took a mere four months to complete and cost only 3x more than our initial high-end estimate. More importantly, interlocutors are happily silent (at least until the next home improvement project begins) .

When asked if I miss being the lead developer of mixed model software at SAS, I usually reply “yes and no.” The “yes” comes from feeling very fortunate to have been involved with this powerful methodology throughout my nearly 20 years at SAS and a part of the strong legacy that began with the work of Jim Goodnight on PROC GLM and PROC VARCOMP in the early days of the company. A big debt of gratitude goes to Dave Delong, who initially pointed me in this direction, and to John Sall and Randy Tobias (our newest ASA Fellow!), who provided key pieces of code that formed the beginnings of PROC MIXED. Interestingly, John’s code was based on the pioneering work of Jennrich and Schluchter on covariance-structure modeling (the R side of the mixed model), while Randy’s code focused on generalizing REML estimation of variance components (the G side of the mixed model). Putting the two together was somewhat serendipitous and turned out to provide a wonderful technology for such diverse applications as animal breeding, clinical trials, manufacturing quality control, education assessment, space-time modeling and statistical genetics.

The “no” comes from confidence in the very impressive work done by our current mixed model gurus: Oliver Schabenberger (PROC GLIMMIX), Chris Gotwalt (JMP), Paul Wright (EVAAS) and Tianlin Wang (PROC HPMIXED). That, and my now decade-long passion for genomics and the true joy that comes from leading a top-notch team and working with all of the dedicated professionals who have helped JMP Genomics become successful.

I’ve been amazed at how mixed model theory is a unifying theme throughout statistics, encompassing such methods as empirical Bayes, ridge regression, time series, kriging, and smoothing splines. Did you know you can fit a support vector machine using the radial basis functions in PROC GLIMMIX? In a bit of a throwback to SAS in the '70s, we’re using variance components on principal components to quantitatively compare sources of variability in microarray data. Most recently, we’ve been exploring how mixed models can effectively adjust for population stratification in genome-wide association studies. So I can’t seem to get away from mixed models — not that I really want to!

Let me close with a mixed model curiosity that I was looking at just last week. Consider the following SAS code, which compares a simple linear regression model with the same model fit with random effects:

/* simulate data for a simple linear regression and compare fixed and random models */
%let seed = 2817340;
data xy;
	do slope = 1 to 5;
		do nx = 1 to 10;
			x1 = rannor(&seed);
			x2 = rannor(&seed);
			do rep = 1 to 10;
				y = x1*slope + x2*slope*2 + rannor(&seed);
				output;
			end;
		end;
	end;
run;

ods exclude all;
ods noresults;
proc mixed data=xy method=ml;
	by slope;
	model y = x1 x2 / s;
	ods output fitstatistics=fs covparms=cp solutionf=sf;
run;
ods exclude none;
ods results;

title "Fit Statistics from Fixed Model";
proc sort data=fs;
	by descr slope;
run;
proc print data=fs;
run;
title "Covariance Parameters from Fixed Model";
proc print data=cp;
run;
title "Beta Hat from Fixed Model";
proc sort data=sf;
	by Effect Slope;
run;
proc print data=sf;
run;

ods exclude all;
ods noresults;
proc mixed data=xy method=ml;
	by slope;
	model y = / s;
	random x1 x2 / s;
	ods output fitstatistics=fsr covparms=cpr solutionf=sfr solutionr=srr ;
run;
ods exclude none;
ods results;

title "Fit Statistics from Random Model";
proc sort data=fsr;
	by descr slope;
run;
proc print data=fsr;
run;
title "Covariance Parameters from Random Model";
proc sort data=cpr;
	by CovParm Slope;
run;
proc print data=cpr;
run;
title "Beta Hat from Random Model";
proc print data=sfr;
run;
title "Gamma Hat from Random Model";
proc sort data=srr;
	by effect slope;
run;
proc print data=srr;
run;

The second PROC MIXED call does something that I would have (up until last week!) recommended against — using simple covariates in a RANDOM statement with no SUBJECT= effect. It turns out that the gamma-hat estimates (the empirical BLUPs) are virtually identical with the beta-hat estimates from the first simple regression model. So covariance structure modeling is intimately connected with mean modeling.

If you’re really into mixed models (or would like to be), here’s a question for you: Why are the variance component estimates from the second model approximately equal to the square of the slopes? Hint: The answer is not “yes and no.”