It's still hyperbolic to assume that two data points (4 and 16 toys) extrapolate to a linear correlation though. The results could just as well support a hypothesis that the correlation with toy engagement is a bell curve peaking somewhere between 4 and 16.
There's an infinite number of functions that can go through two points in cartesian space. What's the simplest? Generally, the simpler the function, the more plausible it is.
I didn't expect that to get downvoted, since it's a mathematical fact. The more degrees of freedom you have in your hypothesis space, the greater the average distance of any particular hypothesis from the correct one. Curse of dimensionality, Occam's razor, etc.
I wasn't the one downvoting, but my guess is that it was related to your claim about plausibility. Just because something is simpler or more plausible it doesn't mean it's more likely to be correct. If it were, there would be no reason to be conducting studies like this
In this case, I did indeed mean more plausible in the sense of higher probability of being correct. If I remembered more of my Machine Learning classes I'd be able to quote you the mathematical proof, but it matches the intuition of Occam's Razor -- models with fewer parameters are, in general, better. You're less likely to overfit, etc.
> If it were, there would be no reason to be conducting studies like this
I don't understand why. The simplest model is to believe there's no relationship (zero parameters). The study gives evidence of a relationship (one parameter). The other commenter was suggesting a parabolic relationship (two parameters), an unnecessary complexity.
The best policy is to believe the simplest model that is consistent with the evidence. If a new study observes the behavior at 6 toys and 8 toys that doesn't line up with the previous study, then I might believe a more complex relationship. Or might go back to believing no relationship.
> The hypothesis that fewer toys in children’s environments would improve the quality of play, measured by three variables, was supported.
But given they only conducted a paired difference test for 4 toys and 16 toys, they can only conclude that there's a statistically significant difference between 4 and 16 toys. This is very different than saying fewer toys correlate to any result, because when one says "fewer toys correlate to X", it implies that the correlation stands for any two numbers of toys, and that's not what the study looked at.
Sure. But if we're going to go that far down the path of pedantry, why not get to the even deeper problem with that statement, which is that, strictly speaking, your alternative hypothesis is never supported; you can only fail to reject the null hypothesis.
I don't think it's that pedantic a point. IMHO, I agree this study is incremental and thus too preliminary to draw generalizing conclusions on. Even the paper itself acknowledges a number of major aspects that further research should focus on.
But consider that threads here are already drawing parallels to Montessori education, personal anecdotes, etc. People are used to big bang results and will happily draw conclusions that aligns with their world view even if the wording in a paper isn't exactly what a scientific study actually means to say.
