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Nested Dichotomy

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The most general type of logit model (see The Paired Comparison Model) is the polytomous logit model. In this model, the response variable has two or more possible options, and the probability of choosing those options is determined from the probability of each but the last one, which is written in terms of the others (since, for instance, the probability of option three out of three options would be anything that’s not the probability of options one or two). This is a good choice for data where you have many options that are not easily broken down into either/or type questions, such as lists of three or more political candidates.

Another type of model is the nested dichotomy. With this approach, you do multiple models based on two-option scenarios (i.e. a yes/no question), which are “nested” within one another as appropriate. (For instance, “Do you work?” would be one level, and then the next, for the answer “yes,” would be, “Full-time or part-time?”) While it is theoretically possible to use this in a great many scenarios, it’s most effective when selected at times where it makes sense—such as the above. In something like a list of five candidates, it would be silly, because then you’d have to ask, “Do you prefer Bob, or someone other than Bob?” and so on.

Nested dichotomy regressions are also handy because due to the way the models are framed, the probabilities of one level are independent of those in the next, so you can just put them together.

***

            “Was it a nightmare, or something else?”

            Sherlock huffed.

            “Okay,” John muttered, and gripped Sherlock’s leg, which was draped over John’s midsection, hoping it might help ground him. “If you say something about nightmares being for the feeble-minded…”

            He huffed again, and shuffled closer to John. John had thus far avoided mentioning the slight shiver that ran though Sherlock’s shoulders every so often.

            “Was it that you had a nightmare at all that’s,” John paused, choosing his words carefully, “concerning you, or was it what was in it?”

            “In retrospect, it was completely infeasible. Even unconscious I should have been able to identify that the footprints were from work boots, not…”

            “So it was that you weren’t as sharp,” John concluded, and Sherlock shook his head. “That you’ve woken up and realized that you can’t trust your mind in your sleep?”

            Sherlock laid his chin on John’s shoulder. “I already knew that,” he said quietly.

            “Right.” John had never heard Sherlock make any noise in his sleep before, but that didn’t mean he never dreamt—medically speaking, it was guaranteed that he dreamt—it just meant that any dreaming, any horrors he experienced at night, were silent. John supposed this wasn’t a surprising night for Sherlock’s mind to conjure up something horrific, anyway. John hadn’t gone to sleep, for similar reasons—had just laid in bed, reading, until Sherlock barged in.

 

 

            Water dripped from the ceiling.

            Water dripped from the ceiling insofar as that water dripped from the ceiling tiles, which were located in pieces on the floor.

            Sherlock stood, brushing dust from himself, brushing the ceiling from himself, and looked beneath him and saw his own unconscious—no, dead—body, on the ceiling.

            Rewind.

            The ceiling gathered back together, coagulated thirty feet above him, spat water back out into the pool as it gathered up tendrils of itself back into the depths. Sherlock felt heat being sucked out of him, felt himself go from fired ceramic to clammy chill heated by adrenaline alone, felt himself get lighter as the ceiling rose, was drawn back into his body as it was born and awoke and soared, heels-first, back to the edge of the pool, and another body shed the ceiling and soared toward him, and John’s arms wrapped around him, and then John let go, and ran farther and farther away, heels-first, and Jim opened the door with his very presence and stepped back through it.

            Play.

            Jim left, and John glanced around, eyes sharp, and must have seen something he didn’t like, because he ran toward Sherlock and clung onto him and leaned toward the pool and almost dumped them both in before the Semtex, which had been thrown into the pool farther down the lane, blew.

            Back, back, back.

            Rewind.

            Sherlock was outside his body again.

            Sherlock knew this was going to happen; Sherlock was up in the rafters behind a crouched figure: the sniper. He climbed down the ladder and he paced back to the outside of the pool, where he bent down and undiscovered the footprints.

            Stop. Step forward frame by frame.

            In the silence Sherlock could nearly hear his knees creak. He watched himself squat and squint at the prints, nails displacing insignificant amounts of dirt as he traced over them. His eyes widened and his pupils adjusted like camera irises as he made a realization. His weight shifted slightly in the dirt, and he left his own prints, too. He took breath into his lungs in slow-motion reverse-explosions of cold evening air rushing into hot lungs, expelling water that swirled through the air like cigarette smoke.

            He stood and stared in the direction of the footprints, the footprints of the sniper.

            Fast forward.

            Sherlock ran into the rafters and dashed up the ladder and came to a screeching halt behind the sniper. Sherlock looked down over the railing and saw himself and John, frozen, even in fast forward.

            Stop.

            Sherlock had gone back to do it right this time. He had found the sniper and he was going to stop him, this time, from setting off the Semtex just when Jim got a safe distance away.

            Below, John and Sherlock were frozen, now by design, John just about to sigh as Jim left, Sherlock looking on, eyes gleaming, glazed over with something he couldn’t see from here but knew anyway, because he had been there, once. In just a few seconds, he would take another breath, and look to John, and John would lock eyes with him and everything would slow down whether Sherlock willed it or not, because that was how it had happened. They would look, and understand, and this time, the bomb wouldn’t go off, because Sherlock would stop it.

            Except he wouldn’t, because it did.

            Play.

            Sherlock leapt onto the sniper, forcing his hands away from his gun, but found them empty.

            Pause.

            John glanced upward toward them and then over to the other side of the rafters.

            Step forward one frame.

            A small flash, a jerking movement.

            Step forward one frame.

            A bullet hovered just outside the barrel of a rifle on the other side of the rafters.

            Play.

            John dashed for Sherlock, holding his arms tight around him. The bomb blew: Sherlock was sucked back into his body.

            John’s grip was fierce and tight.

            Rewind. Stop. Play.

            John started running.

           The bullet whirred through the air, punctured the surface of the water.

           John wrapped his arms around Sherlock until the force of the explosion tore him away, and he and Sherlock were sent flying back away from each other in the blazing heat, the ceiling crashing down upon them.

            Sherlock stood and brushed dust from himself as water dripped from the ceiling tiles onto his shoes. His body lay beneath him. He felt a tugging toward the northeast wall.

            Pause.

            He couldn’t turn back.

            Pause.

            Rewind.

            Pause.

            He couldn’t go back.

            Play.

            Sherlock approached the wall—what remained of it—cautiously, stepping around the rubble as if he were anything more than a projection, a ghost, a spirit, as if he could displace them if he tried.

            He could displace them if he tried.

            Pause. Stop. Stop, stop, stop.

            Sherlock crouched over the rubble. He felt a tugging.

            He pulled back one fragment of the wall, and then another, and then a bit of the ceiling, and then a light fixture which had managed to burn into flesh.

            Stop, stop, stop, stop.

            The flesh was John’s.

            Stop, stop, stop, rewind, stop.

            Sherlock peeled away each layer of rubble like clothing, like he’d done the coat with the bomb just minutes ago.

            Stop.

            John lay bruised and burned and broken beneath them, and Sherlock laid a hand to his face, and it was still warm, and he leaned down to breathe life back into John but he couldn’t, because the world stopped then, everything in the world but his mind, with his mouth pressed over John’s, with one hand holding John’s sticky-burnt fingers inside it, with one hand pressed along the contours of John’s face, thumb nudging at his cheek, tips of his fingers just touching singed hair, just feeling bruising that would deepen, if the world would un-pause.

            Sherlock could still think, though, could still think about that he could feel John’s bones too well for his hand to be intact, could think about infections and the dust from the ceiling and the wound on John’s head, could think about the stillness of John’s chest, housing no slow-motion reverse-explosions, expelling no cigarette-smoke moisture into the chill brought on by the lack of walls and ceiling. He could think, but in the end, his thoughts spiraled down to two things, and there he was stuck: John, dead, John, dead, John, dead.

            Sherlock woke, gasping.

 

 

            He had entered John’s room exactly as he’d gotten out of bed.

            “Sherlock?” John had glanced up from his book and, seeing the state of him, quickly snapped the book shut, turned off the light for the sake of some sense of propriety. “What’s wrong?”

            Sherlock had approached the bed and threw back the blankets and collapsed onto it and curled around John, wrapping one arm around his shoulder, wrapping the other around his chest, wrapping his leg around John’s belly and hips, measuring the rate at which John breathed in and breathed out, which was faster than not at all, which was perfect—John heard him mutter as much. “Like the opposite of an explosion,” Sherlock murmured, and nuzzled his head up against John’s shoulder.

            “Are you okay?”

            Sherlock finally glanced up at John, wrapped himself up tighter around him. John was too preoccupied with the quickness of Sherlock’s breathing, with the fervor with which he clung to John—too preoccupied with worry, really—to do anything other than note to himself that Sherlock was naked, and so was he, and they were touching in places John had never really quite gotten around to imagining them touching, and maybe not at all in the way he would have imagined it happening, either. He looked back to Sherlock’s eyes, wide and observing John’s face, drinking him in, convincing himself of something or another about John, probably somehow related to whatever had happened.

            It could be anything—but it was probably a nightmare. John knew one when he saw one.

            “It’s all right,” he muttered, and tentatively laid his hand on Sherlock’s leg. “It’s okay now.”

            “No,” Sherlock’s eyes scanned John, “no. It’s not.” He buried his nose against John’s shoulder.

            “Was it a nightmare, or something else?”

            But that wasn’t the right thing to say, was it? It was a nightmare—obvious. Maybe Sherlock needed a laugh. Maybe that would drag him out of it. John gripped his leg reassuringly, made some crack about only the feeble-minded having nightmares…to no avail.

            So, no diversions, then. Straight to the heart of the matter. “Was it that you had a nightmare at all that’s,” he couldn’t say got you worked up or got you in this state, because that was too loaded, wasn’t it? “…Concerning you, or was it what was in it?”

            “In retrospect,” Sherlock muttered now, “it was completely infeasible. Even unconscious I should have been able to identify that the footprints were from work boots, not…”

            “So it was that you weren’t as sharp.” No, no, apparently not. John took another guess. “That you’ve woken up and realized that you can’t trust your mind in your sleep?”

            Sherlock lifted his nose so that he could rest his chin against John’s shoulder. “I already knew that,” he said quietly.

            “Right.”

            “Possibly related to increased dimethyltryptamine levels,” Sherlock continued, going on about nearly dying and neurotransmitters and N-methyltransferase, and John curled his arm around Sherlock’s neck to stroke his hair, waiting for him to run out of words, which would have taken a while except that while John was nearly drifting off himself, Sherlock leaned forward and pressed his mouth to John’s.

           He held it there for—god—John had no idea, but he didn’t dare pull away. Sherlock’s arm still rested across his chest, so John continued taking deep breaths, since that seemed to comfort him, and grasped Sherlock’s leg firmly, and ran his fingers through the hairs at the nape of Sherlock’s neck.

            After what had to have been several minutes, Sherlock pulled away, and as he did so, his fluttered closed and then stayed closed and then flew open, and he apparently found something about the experience amusing, because he smiled and nuzzled back into John’s shoulder. “Pause,” he said, and, “Step forward frame by frame.”

            John turned toward him and kissed his forehead slowly, lingeringly, avoiding the temptation to try to make sense of his words. “Did we die? In your nightmare?”

            “Yes,” Sherlock whispered, and leaned forward so that his forehead remained against John’s lips. “But it’s okay now.”