I have data of 6 groups with sample size of n = 2, 10, 2, 9, 3, 1 and I want to perform Permutational multivariate analysis of variance (PERMANOVA) on these data.
My question is: Is it correct to run perMANOVA on these data with the small sample size? The results look strange for me because the group of n = 1 showed insignificant difference to other groups although the graphical representation of the groups clearly show a difference.
Thank you
I would not trust any result with group of n=1 because there is no source of variation to define difference among groups.
I have also received some answers from other platforms. I put them here for information:
The sample size is simply too small to yield a stable solution via manova. Note that the n = 1 cell contributes a constant value for that cell's mean, no matter what you do by way of permutations.
Finally, note that the effective per-cell sample size with unequal cell n for one-way designs tracks well to the harmonic mean of n. For your data set as it stands, that means an "effective" per-cell n of about 2.4. Unless differences are gigantic on the DV set, no procedure (parametric or exact/permutation) will have the statistical power to detect differences with that size.
MANOVA emphasizes the attribute scattering in the study group and the logic of this analysis is based on the scattering of scores. It is not recommended to use small groups with one or more people (I mean less than 20 people) to perform parametric tests such as MANOVA. In my opinion, use non-parametric tests to examine small groups.
Related
I want to run regression spline with B-spline basis function. The data is structured in such a way that the number of observations is less than the number of basis functions and I get a good result.
But I`m not sure if this is the correct case.
Do I have to have more rows than columns like linear regression?
Thank you.
When the number of observations, N, is small, it’s easy to fit a model with basis functions with low square error. If you have more basis functions than observations, then you could have 0 residuals (perfect fit to the data). But that is not to be trusted because it may not be representative of more data points. So yes, you want to have more observations than you do columns. Mathematically, you cannot properly estimate more than N columns because of collinearity. For a rule of thumb, 15 - 20 observations are usually needed for each additional variable / spline.
But, this isn't always the case, such as in genetics when we have hundreds of thousands of potential variables and small sample size. In that case, we turn to tools that help with a small sample size, such as cross validation and bootstrap.
Bootstrap (ie resample with replacement) your datapoints and refit splines many times (100 will probably do). Then you average the splines and use these as the final spline functions. Or you could do cross validation, where you train on a smaller dataset (70%) and then test it on the remaining dataset.
In the functional data analysis framework, there are packages in R that construct and fit spline bases (such as cubic, B, etc). These packages include refund, fda, and fda.usc.
For example,
B <- smooth.construct.cc.smooth.spec(object = list(term = "day.t", bs.dim = 12, fixed = FALSE, dim = 1, p.order = NA, by = NA),data = list(day.t = 200:320), knots = list())
constructs a B spline basis of dimension 12 (over time, day.t), but you can also use these packages to help choose a basis dimension.
No idea if I am asking this question in the right place, but here goes...
I have a set of equations that were calculated based on numbers ranging from 4 to 8. So an equation for when this number is 5, one for when it is 6, one for when it is 7, etc. These equations were determined from graphing a best fit line to data points in a Google Sheet graph. Here is an example of a graph...
Example...
When the number is between 6 and 6.9, this equation is used: windGust6to7 = -29.2 + (17.7 * log(windSpeed))
When the number is between 7 and 7.9, this equation is used: windGust7to8 = -70.0 + (30.8 * log(windSpeed))
I am using these equations to create an image in python, but the image is too choppy since each equation covers a range from x to x.9. In order to smooth this image out and make it more accurate, I really would need an equation for every 0.1 change in number. So an equation for 6, a different equation for 6.1, one for 6.2, etc.
Here is an example output image that is created using the current equations:
So my question is: Is there a way to find the relationship between the two example equations I gave above in order to use that to create a smoother looking image?
This is not about logarithms; for the purposes of this derivation, log(windspeed) is a constant term. Rather, you're trying to find a fit for your mapping:
6 (-29.2, 17.7)
7 (-70.0, 30.8)
...
... and all of the other numbers you have already. You need to determine two basic search paramteres:
(1) Where in each range is your function an exact fit? For instance, for the first one, is it exactly correct at 6.0, 6.5, 7.0, or elsewhere? Change the left-hand column to reflect that point.
(2) What sort of fit do you want? You are basically fitting a pair of parameterized equations, one for each coefficient:
x y x y
6 -29.2 6 17.7
7 -70.0 7 30.8
For each of these, you want to find the coefficients of a good matching function. This is a large field of statistical and algebraic study. Since you have four ranges, you will have four points for each function. It is straightforward to fit a cubic equation to each set of points in Cartesian space. However, the resulting function may not be as smooth as you like; in such a case, you may well find that a 4th- or 5th- degree function fits better, or perhaps something exponential, depending on the actual distribution of your points.
You need to work with your own problem objectives and do a little more research into function fitting. Once you determine the desired characteristics, look into scikit for fitting functions to do the heavy computational work for you.
I conducted an Exploratory Factor Analysis (Principal Axis Factoring) on my data and wanted to determine the number of factors to extract via. Horn's Parallel Analysis.
However I have two problems:
The parallel analysis suggests to extract 1 factor, however the plot shows more than one intersection of my "FA Actual Data" and my "FA Simulated Data" line. I do not get why it is just one factor (the first intersection) then.... This plot does not look typical to other parallel analysis plots.
Why does the number of factors to extract change with the number of observations (n.obs) I state? I mean that I just changed the number of observations from 50 to 500 (which is a lie), however then parallel analysis suggested 5 factors to extract instead of 9. I do not get why....
Thank you so much for any helpful tips.
Valerie
fa.parallel(cor(My_Data), n.obs = 50, fa="fa", fm="pa")
Parallel analysis suggests that the number of factors = 1 and the number of components = NA
These questions regard a set of data with lists of tasks performed in succession and the total time required to complete them. I've been wondering whether it would be possible to determine useful things about the tasks' lengths, either as they are or with some initial guesstimation based on appropriate domain knowledge. I've come to think graph theory would be the way to approach this problem in the abstract, and have a decent basic grasp of the stuff, but I'm unable to know for certain whether I'm on the right track. Furthermore, I think it's a pretty interesting question to crack. So here we go:
Is it possible to determine the weights of edges in a directed weighted graph, given a list of walks in that graph with the lengths (summed weights) of said walks? I recognize the amount and quality of permutations on the routes taken by the walks will dictate the quality of any possible answer, but let's assume all possible walks and their lengths are given. If a definite answer isn't possible, what kind of things can be concluded about the graph? How would you arrive at those conclusions?
What if there were several similar walks with possibly differing lengths given? Can you calculate a decent average (or other illustrative measure) for each edge, given enough permutations on different routes to take? How will discounting some permutations from the available data set affect the calculation's accuracy?
Finally, what if you had a set of initial guesses as to the weights and had to refine those using the walks given? Would that improve upon your guesstimation ability, and how could you apply the extra information?
EDIT: Clarification on the difficulties of a plain linear algebraic approach. Consider the following set of walks:
a = 5
b = 4
b + c = 5
a + b + c = 8
A matrix equation with these values is unsolvable, but we'd still like to estimate the terms. There might be some helpful initial data available, such as in scenario 3, and in any case we can apply knowledge of the real world - such as that the length of a task can't be negative. I'd like to know if you have ideas on how to ensure we get reasonable estimations and that we also know what we don't know - eg. when there's not enough data to tell a from b.
Seems like an application of linear algebra.
You have a set of linear equations which you need to solve. The variables being the lengths of the tasks (or edge weights).
For instance if the tasks lengths were t1, t2, t3 for 3 tasks.
And you are given
t1 + t2 = 2 (task 1 and 2 take 2 hours)
t1 + t2 + t3 = 7 (all 3 tasks take 7 hours)
t2 + t3 = 6 (tasks 2 and 3 take 6 hours)
Solving gives t1 = 1, t2 = 1, t3 = 5.
You can use any linear algebra techniques (for eg: http://en.wikipedia.org/wiki/Gaussian_elimination) to solve these, which will tell you if there is a unique solution, no solution or an infinite number of solutions (no other possibilities are possible).
If you find that the linear equations do not have a solution, you can try adding a very small random number to some of the task weights/coefficients of the matrix and try solving it again. (I believe falls under Perturbation Theory). Matrices are notorious for radically changing behavior with small changes in the values, so this will likely give you an approximate answer reasonably quickly.
Or maybe you can try introducing some 'slack' task in each walk (i.e add more variables) and try to pick the solution to the new equations where the slack tasks satisfy some linear constraints (like 0 < s_i < 0.0001 and minimize sum of s_i), using Linear Programming Techniques.
Assume you have an unlimited number of arbitrary characters to represent each edge. (a,b,c,d etc)
w is a list of all the walks, in the form of 0,a,b,c,d,e etc. (the 0 will be explained later.)
i = 1
if #w[i] ~= 1 then
replace w[2] with the LENGTH of w[i], minus all other values in w.
repeat forever.
Example:
0,a,b,c,d,e 50
0,a,c,b,e 20
0,c,e 10
So:
a is the first. Replace all instances of "a" with 50, -b,-c,-d,-e.
New data:
50, 50
50,-b,-d, 20
0,c,e 10
And, repeat until one value is left, and you finish! Alternatively, the first number can simply be subtracted from the length of each walk.
I'd forget about graphs and treat lists of tasks as vectors - every task represented as a component with value equal to it's cost (time to complete in this case.
In tasks are in different orderes initially, that's where to use domain knowledge to bring them to a cannonical form and assign multipliers if domain knowledge tells you that the ratio of costs will be synstantially influenced by ordering / timing. Timing is implicit initial ordering but you may have to make a function of time just for adjustment factors (say drivingat lunch time vs driving at midnight). Function might be tabular/discrete. In general it's always much easier to evaluate ratios and relative biases (hardnes of doing something). You may need a functional language to do repeated rewrites of your vectors till there's nothing more that romain knowledge and rules can change.
With cannonical vectors consider just presence and absence of task (just 0|1 for this iteratioon) and look for minimal diffs - single task diffs first - that will provide estimates which small number of variables. Keep doing this recursively, be ready to back track and have a heuristing rule for goodness or quality of estimates so far. Keep track of good "rounds" that you backtraced from.
When you reach minimal irreducible state - dan't many any more diffs - all vectors have the same remaining tasks then you can do some basic statistics like variance, mean, median and look for big outliers and ways to improve initial domain knowledge based estimates that lead to cannonical form. If you finsd a lot of them and can infer new rules, take them in and start the whole process from start.
Yes, this can cost a lot :-)
What is the best way to constrain the values of a PRNG to a smaller range? If you use modulus and the old max number is not evenly divisible by the new max number you bias toward the 0 through (old_max - new_max - 1). I assume the best way would be something like this (this is floating point, not integer math)
random_num = PRNG() / max_orginal_range * max_smaller_range
But something in my gut makes me question that method (maybe floating point implementation and representation differences?).
The random number generator will produce consistent results across hardware and software platforms, and the constraint needs to as well.
I was right to doubt the pseudocode above (but not for the reasons I was thinking). MichaelGG's answer got me thinking about the problem in a different way. I can model it using smaller numbers and test every outcome. So, let's assume we have a PRNG that produces a random number between 0 and 31 and you want the smaller range to be 0 to 9. If you use modulus you bias toward 0, 1, 2, and 3. If you use the pseudocode above you bias toward 0, 2, 5, and 7. I don't think there can be a good way to map one set into the other. The best that I have come up with so far is to regenerate the random numbers that are greater than old_max/new_max, but that has deep problems as well (reducing the period, time to generate new numbers until one is in the right range, etc.).
I think I may have naively approached this problem. It may be time to start some serious research into the literature (someone has to have tackled this before).
I know this might not be a particularly helpful answer, but I think the best way would be to conceive of a few different methods, then trying them out a few million times, and check the result sets.
When in doubt, try it yourself.
EDIT
It should be noted that many languages (like C#) have built in limiting in their functions
int maximumvalue = 20;
Random rand = new Random();
rand.Next(maximumvalue);
And whenever possible, you should use those rather than any code you would write yourself. Don't Reinvent The Wheel.
This problem is akin to rolling a k-sided die given only a p-sided die, without wasting randomness.
In this sense, by Lemma 3 in "Simulating a dice with a dice" by B. Kloeckner, this waste is inevitable unless "every prime number dividing k also divides p". Thus, for example, if p is a power of 2 (and any block of random bits is the same as rolling a die with a power of 2 number of faces) and k has prime factors other than 2, the best you can do is get arbitrarily close to no waste of randomness, such as by batching multiple rolls of the p-sided die until p^n is "close enough" to a power of k.
Let me also go over some of your concerns about regenerating random numbers:
"Reducing the period": Besides batching of bits, this concern can be dealt with in several ways:
Use a PRNG with a bigger "period" (maximum cycle length).
Add a Bays–Durham shuffle to the PRNG's implementation.
Use a "true" random number generator; this is not trivial.
Employ randomness extraction, which is discussed in Devroye and Gravel 2015-2020 and in my Note on Randomness Extraction. However, randomness extraction is pretty involved.
Ignore the problem, especially if it isn't a security application or serious simulation.
"Time to generate new numbers until one is in the right range": If you want unbiased random numbers, then any algorithm that does so will generally have to run forever in the worst case. Again, by Lemma 3, the algorithm will run forever in the worst case unless "every prime number dividing k also divides p", which is not the case if, say, k is 10 and p is 32.
See also the question: How to generate a random integer in the range [0,n] from a stream of random bits without wasting bits?, especially my answer there.
If PRNG() is generating uniformly distributed random numbers then the above looks good. In fact (if you want to scale the mean etc.) the above should be fine for all purposes. I guess you need to ask what the error associated with the original PRNG() is, and whether further manipulating will add to that substantially.
If in doubt, generate an appropriately sized sample set, and look at the results in Excel or similar (to check your mean / std.dev etc. for what you'd expect)
If you have access to a PRNG function (say, random()) that'll generate numbers in the range 0 <= x < 1, can you not just do:
random_num = (int) (random() * max_range);
to give you numbers in the range 0 to max_range?
Here's how the CLR's Random class works when limited (as per Reflector):
long num = maxValue - minValue;
if (num <= 0x7fffffffL) {
return (((int) (this.Sample() * num)) + minValue);
}
return (((int) ((long) (this.GetSampleForLargeRange() * num))) + minValue);
Even if you're given a positive int, it's not hard to get it to a double. Just multiply the random int by (1/maxint). Going from a 32-bit int to a double should provide adequate precision. (I haven't actually tested a PRNG like this, so I might be missing something with floats.)
Psuedo random number generators are essentially producing a random series of 1s and 0s, which when appended to each other, are an infinitely large number in base two. each time you consume a bit from you're prng, you are dividing that number by two and keeping the modulus. You can do this forever without wasting a single bit.
If you need a number in the range [0, N), then you need the same, but instead of base two, you need base N. It's basically trivial to convert the bases. Consume the number of bits you need, return the remainder of those bits back to your prng to be used next time a number is needed.