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You may want to do an analysis of variance using two factors instead of one—a two-way ANOVA, instead of one-way (see Between Variation). We do this by dividing our factors into a row and a column. (For instance, say you wanted to consider whether you were analyzing income by gender and type of job. You would, say, have rows for the job types, and columns for the genders.)
With two variables, it’s always possible that the two will interact with one another. So first, we want to look at what to expect if the variables don’t interact so we can tell if they do interact. If there’s no interaction between the variables, then the difference between averages of different columns for the same row variable would all equal each other (or vice versa, the difference between averages for different rows for the same column variable would equal each other). We can visualize this using a plot. Consider the following diagram:
The red line and the blue line represent two different rows of information, whereas the C’s on the x-axis are two different columns. So notice for this data that both of the lines rise by the same amount from point “1” to point “2” (brackets on the right). This means that there are “column effects”—changing from column 1 to column 2 has an effect. But since the changes are the same, we have a case for saying there is no interaction. Then, look at the fact that the difference in height between the two different row values at C1 is equal to the height difference at C2 (brackets on the left). There are “row effects,” because different rows have different values. If the lines were right on top of one another, there would not be row effects. But since the changes are the same here, too, then we can say that this graph shows no interaction between the two variables (rows and columns). Another intuitive way to look at it is to check if they are parallel to one another: this would mean that the difference between the two vertically is always the same so there is no interaction. The next diagram is an example of “disordinal interaction”—not only is there interaction here, but it is called “disordinal” because the values of the rows “switch places.”
So here you can see that the lines are definitely not parallel. The difference between the two points changes from column to column and from row to row.
It is also important to note that when there isn’t interaction, the effects of the two factors are additive. So if you want to compare, say, the first point of the first column to the second point of the second column, you just have to add the mean change in column (the amount point 2 from data set 1 is up or down from point 1 in data set 1) to the change in row (how far point 1 from data set 1 is from point 1 in data set 2).
In general, the first thing we want to check for is whether there is any interaction. We would set up a test with the hypothesis that there isn’t any, and then look to see if we can find some. This is just so that we know what we’re dealing with, though, and not the overall goal. What we really want is the “main effects”—the row effects and the column effects. Our hypotheses would be that there are no row or column effects, respectively (what this would look like on the above graphs would be two horizontal lines right on top of one another!), with the alternative being that there are row or column effects, and we test to see which is true. Whether or not there are interactions involved will impact the model we use in doing so.
***
While John and Sherlock were apart (apart for years, this time, years and years and years even with both in London, apart but for the occasional visits, ever less personal) they were not so different.
Sherlock still solved crimes.
He pretended to shoot holes in the wall. It was better this way.
It wasn’t better this way, of course, under various definitions of it. (His heart, he thought, but didn’t think think, didn’t bring to the surface, just let sit there under the film that formed over that part of his mind that he hoped one day would become a skin or a wall, if it came to that, if John didn’t come back to burst through it, because Sherlock sure as hell would not be doing it himself. His heart was not better this way. It was much, much worse. John was gone. John lived less than eight miles away. John was gone.)
But what was better this way was the wall, and at least Mrs. Hudson appreciated that. Sherlock knew better than to push his luck without John here to diffuse the tension, but he also knew that Mrs. Hudson would never throw him out of 221B. Mrs. Hudson understood. She knew what Sherlock would do to himself without a place to tie him down, without a not-housekeeper to pretend that he might occasionally have some tea and biscuits with her and to fret over the state of the kitchen and to dust up things that Sherlock didn’t want dusted—even if she didn’t know what Sherlock would do to himself without her, without 221B.
She had been there for all of it, of course. When Sherlock and John had—well. When they had stumbled into the flat and started snogging before they even got to their door, when Mrs. Hudson poked her nose into the flat the next morning to find them piled on the couch, John’s head buried in the crook of Sherlock’s neck as he slept in naught but his pants, the red and the purple marking Sherlock’s exposed leg and neck and everywhere else that Mrs. Hudson could see from her vantage point. Sherlock had cracked an eye open and smiled, and she had giggled and left, and come back later with fancy biscuits and Sherlock left John to wonder how exactly she knew. “I always knew it,” she’d said. “Oh, I saw it in your eyes, boys, I just knew it would happen one day.” And neither of them could be bothered to argue, nestled up in 221B with their favorite person in the world, and their favorite landlady, too.
And then, of course, had come the—the ending of it. “I just know you two will patch it up,” Mrs. Hudson had said, once or twice, when she had caught Sherlock staring off into the distance, dazed and lost.
When John got married and left, Mrs. Hudson congratulated him, and went to the wedding, and everything, but Sherlock knew—well, perhaps he only wanted to see—but he thought he could see a hesitation too, because she knew the wrongness of Sherlock-and-not-John at 221B. She still, bless her, even while wishing John the best and meaning it, because she had grown so fond of him and been there for him when Sherlock had died, she still held out hope that John would come to his senses and come back. Or so Sherlock liked to believe, in any case.
When John was gone, Mrs. Hudson checked on Sherlock more. He indulged her in more chats over tea than he ever thought he could bear. Mostly they consisted of gossip: Mrs. Hudson would talk about Mrs. Turner, and then goad Sherlock into letting loose all the nasty things about people he’d solved cases before. She would pepper his explanations with, “Sherlock, that’s rather rude, isn’t it?” or “Do you really think you ought to say that aloud?” but she’d asked, and she enjoyed it, Sherlock knew. Even if he’d turned on her, spouted everything he knew about anything questionable she’d ever done, she’d listen, because Mrs. Hudson just liked the idea of Sherlock talking, of being out in the world and not in his head all the time. And she—she was lonely too.
It had been bad the one-year anniversary of John’s marriage to Mary; it had been worse on his fifth. (One could be mere chance. Not nearly enough time for things to break down. Five, and still going strong: that was worse, far worse.) Sherlock had lit a candle. He locked himself in the bathroom with a filling tub and no idea if he was just going to wash off or fall asleep there long past the candle’s reaching its end, and past and past and past. He had considered shooting up but decided he was better than that, would not give anyone (John, John John John) the satisfaction. He had considered shooting up just a little too much and calling John and saying it was an accident and pleading for help, come soon, if only you were here right now; was an apparent loss of self-control worth the guilt he could inflict? Worth John being there, kneeling over him in the bath, candle burning low, on his own anniversary?
Of course it was stupid. Of course it was completely stupid. Sherlock watched the bath fill and debated whether he really wanted to wash off. There was probably an experiment somewhere on his list that he could do in the bathtub. Something nasty that John never would have stood for. Something John would have kicked Sherlock onto the sofa for for a week. Slime mold? No, too mundane. Blood stains? He already had sufficient data for most cases. Maybe he could pile up sheep livers in the water and see how long they had to sit in it before they deteriorated into stew. Spiteful, completely useless. His mobile rang and it was Mycroft. Sherlock threw it into the bathtub.
“Sherlock?” Mrs. Hudson had called from the door, knocking lightly. “Hoo-hoo,” she let loose her typical call. “You in there, Sherlock?”
“Yes,” he had finally said.
“Are you decent? May I come in?”
“I’m never decent, Mrs. Hudson,” he said, opening the door. He was, after all, still wearing his usual suit pants and dress shirt.
“What’s going on in here?” she asked. “I hope you’re not going to ruin the bathtub.”
“Of course not,” Sherlock said, and leaned against the counter. He stared at the candle a foot to his left.
“Sherlock, dearie, let’s have dinner together. I won’t go about pretending there’s no reason for you to be upset today.” She reached around him for the candle and put it out. “But that’s no reason for you to be miserable by yourself all evening.”
“Not hungry,” Sherlock said as she bustled over to the tub to turn the faucets off and unplug the drain.
In the end, she had gotten him out of the flat, and they went to a little restaurant with almost lethally rich desserts—Mycroft’s sort of place, Sherlock thought, and thought about his phone in the bottom of the empty bathtub. He’d have to replace it tomorrow. He could change his number—but then, everyone knew it from his website, it would be a terrible inconvenience to lose cases to people being too lazy to look up the revised number and—
And no, that wasn’t the reason, of course. If John called—well. If John called, he wanted to know. They still met, sometimes. They still solved cases. It was nothing like it had been. It was years and years and years since the last time he and John had—
Sherlock pretended to shoot holes in the wall.
John pretended to shoot holes in the wall.
Things hadn’t changed so much. He still solved cases with Sherlock, sometimes.
He pretended to shoot holes in the wall.
It was better this way. It, of course, could be any number of things: Sherlock with no one to complain about his experiments. John, getting regular sleep. The imaginary holes in the wall, for being imaginary and for being in the wall, rather than anywhere else.
He still solved cases with Sherlock sometimes, but he missed the danger when he wasn’t on a case. Or, sometimes, when he was—because something had changed, between them, and of course it sodding had. Pauses when John might have shoved Sherlock up against the wall of the narrow alley, pressed his knee between Sherlock’s legs, a different kind of danger—but instead, he turned away, pretending to be distracted by some sound that would never distract him, distant footsteps or the door to a shop swinging shut.
Aside from not almost dying every other week, though, Mary was exactly, well—well. Perfect. And she knew, bless her, she knew. She’d bring out a plate of biscuits and sit down and they’d have a little chat, and just a few of those times, she acknowledged—it. “I know you miss him,” she said quietly.
“It’s better this way,” John said. “However—however much I miss some things about him, he put me through a lot of shit.”
“I know.”
“I forgot how much I’d missed watching bad telly and not being told two minutes in who the murderer was or what the disease is or that so-and-so’s wife is going to cheat on him sixteen minutes in.”
They had talked about children, off and on, but it hadn’t happened yet. John was fine with it—he felt like he was waiting for something, anyway. Mary was fine with it. She was fine with no kids at all. John was relieved.
Mary was good for him.
Exactly a year after he’d broken up with Sherlock, he’d still lived with him. It had gone completely unmentioned, except for Sherlock’s knowing eyes on his, his legs curled tighter than usual. Six text messages from Mycroft. John had read them as they arrived: 1:05pm, 1:06pm, 1:32pm, 5:49pm, 5:53pm, 7:12pm. Sherlock hadn’t cared as he picked up the phone and scrolled through them.
I’ve something for you to look into, needs attention as soon as possible.
I meant now, Sherlock.
Don’t be overdramatic, brother.
May come by after dinner to bring case file.
Better yet, I shall send a car to retrieve you. Far too busy to cross London now.
Get in the car, Sherlock Holmes.
Sherlock hadn’t, of course. John had heard his phone indicate the arrival of another text message half an hour later, but Sherlock was texting Lestrade at the time, and tucked his mobile in his pocket after that. John read the text the next day. Another from Mycroft.
I could arrange to have John picked up.
Not necessary. Now who’s being overdramatic? –SH
Five years after he’d broken up with Sherlock, John had it rougher, but Mary was good for him, and Mary helped. She helped even though it had been five years, bless her soul, and, John thought, he had no right to still be a wreck over it, but he was. “It’s okay to be upset about it,” she said, rubbing circles into his back. “You loved him very much. That took a great deal of strength to do. It wasn’t bad, John. It was reasonable. You couldn’t just let yourself get hurt over and over.”
John buried his face into his hands. If he would’ve been kinder about it, somehow, if he would have made more of an effort to stay close before he moved out to live with Mary, would they have drifted so far apart? Sherlock was his friend. His best friend. He still was. But things were awful, now. They solved cases together, yes. And then they avoided looking at one another and went their separate ways and John couldn’t imagine if Sherlock went back to life as usual or if he, like John, arrived home and spent twenty minutes staring into the bathroom mirror, wondering if it was usual to have an existential crisis every time one met up with his friend, with his ex-boyfriend. Was it too unhealthy to wonder if he was more of a person when he was with Sherlock than he was now, if he had changed and this was his fault, all of it, and he should have just put up with the thousand ways Sherlock drove him up the wall, all the things not good that left John feeling like crap but at least it went away a day or two later, all but ghosts that haunted him at the worst of times? (Haunted him like now: all the little things Sherlock did wrong, all the little things John did wrong. They could have talked. They could have sorted it out.)
Had Sherlock changed? Was Sherlock less like John was? No, Sherlock could never be less than Sherlock. Did Sherlock ever wonder if it was his fault? No, no, certainly not—he didn’t even seem to remember why it had happened in the first place.
John had been midway through doing the dishes, five years later, before Mary herded him to the sofa to rub circles on his back and reassure him. Harry had been trying to be a good sister, and texting him reassurances, I hurt for ages after Clara, except she and Clara were back together, and Harry was eight months sober. some people love more than one person, John, maybe that’s you, you should and then never mind and John threw his phone into the sink. Yes, yes, yes he still sodding loved Sherlock Holmes and yes, dammit, yes, he loved Mary Morstan, but he couldn’t have both, could he, and Sherlock was bad for him, and Mary was good for him.
He could let his hands wrinkle in the sink, here, for a while. Wait until the water cooled down, feel the chill up his hands and in his arms and watch his fingers and hands grow older and older and older.
“John,” Mary crept up behind him, and slid a hand up onto his shoulder. “Need help finishing those?”
“I’ve got it,” he said quietly, but she helped him anyway, and then pushed him over to the sofa and rubbed circles on his back.
“Want to go get a bite of dessert?” she asked. “I haven’t had cheesecake in ages.”
“Then let’s get some cheesecake,” John said, and they had wine and cheesecake and John tried not to think about Harry still texting away what were supposed to be comforting words, his phone on the bottom of the sink. He’d have to replace it—it’d have to be the same number. For his patients, and all that, and for—well. In case Sherlock ever had to—if it was an emergency and—better safe than sorry. He was a bit fond of the number, anyway—same number he’d had with the mobile he’d gotten from Harry, that he’d finally had to replace six and a half years ago after he had to fish Sherlock out of the Thames. Sherlock had pulled the culprit’s scarf from the pocket of his heavy coat even as John tried to strip the coat off of him, and Lestrade took it, the damning evidence, and John and Sherlock had gone home and John had kissed him and kissed him and—
They still met, sometimes. They still solved cases. It was nothing like it had been. It was years and years and years since the last time he and Sherlock had—
John pretended to shoot holes in the wall.