#stem mondays

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What in the World is ANOVA? Analysis of Variance

by Jared DiDomenico, CAPS Statistics Tutor

What on Earth is ANOVA?

One of the more advanced topics that undergraduate students may start to encounter is the analysis of variance (ANOVA) statistical procedure. Usually students are pretty apt at understanding the t-tests and their procedures, but ANOVA is a bit different. To put it simply, it’s when you want to compare the means of two samples and see if they differ. But what happens if you have more than two groups? What if we are curious about more than one type of factor (e.g. sex, education, etc.)? Then we use the ANOVA model.

Wikipedia defines ANOVA as, “… a collection of statistical models used to analyze the differences among group means and their associated procedures (such as "variation" among and between groups).” That’s quite a mouthful. Essentially, it means we can examine the means of various groups by comparing the variation that we could expect to find naturally between the groups and within the groups. Let’s think about that.

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Variation is something that is always going to be a part of data that consists of random variables. We are all familiar with the normal distribution. Height is an example of something that is normally distributed. This is saying that people in a population will have different heights at various frequencies (i.e. tall people occur less often than people of average height) that can be distributed like a bell curve. If we wanted to examine something like height from an ANOVA point of view it would kind of look like this:

EXAMPLE #1

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Are the heights between the various university sports teams different from one another? Let’s examine the heights of all the soccer, basketball, and football players at UNM (for the sake of this example, data will be made up). We found that the average heights of the soccer players were 70 inches, the football players were 72 inches, and the basketball players were 75 inches. Oh look, they aren’t equal therefore they are different! If anyone has had any statistics classes they know that this isn’t necessarily true because there is natural variation that may have just made our samples different. ANOVA is aimed at seeing if the means of heights in these groups are different and this can be understood if the variation that is present within the soccer players, within the football players, and within the basketball players is different than the variation between the all of the players.

Are the Means Different?

Like any statistical test, we have to have a null hypothesis and an alternative hypothesis. The null hypothesis the hypothesis we assume to be true. Null is usually synonymous with nothing, thus the null hypothesis is no effect. An alternative hypothesis is something that is different from null and cannot occur if the null is true. If the null is no effect, then something mutually exclusive is that there is an effect. Both can’t happen at the same time.

Now, specifically for ANOVA, our null hypothesis is all the group means are the same, or no difference.

The alternative is the opposite, or the compliment in probability, that at least one mean differs from the others. The easiest notation for this is:

To calculate whether or not there are any differences you need to know a few things…

Sums of Squares

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Sums of Squares are a measure of variability within our data. The total sums of squares is a measure of all the variability that is present within our model. (Note: I am going to start calling this ANOVA a model, because that’s what it is, its predicting a response knowing information from what group the response was in, plus a little error). This is a simple, yet tedious calculation, and is as follows:

SSTO (total sums of squares)

Essentially this is saying take the ith value from jth group, subtract the grand mean of the responses from it, square it, and then add it all up using every point in every group.

If we were do some mathematical proofs, we could figure out how to split up the total variation to show the variation between the groups and the variation within the groups. Essentially they are calculated as follows:

SSTR (treatment sums of squares,sometimes called between)

SSE (sums of squares of error, sometimes called within)

This shows us that, SSTO = SSTR +SSE. So it will be easier to solve two of them then use that information to solve the other. At the end of this I will include an example to illustrate the calculations.

The ANOVA Table

We can organize all the needed information by creating an ANOVA table. The general One-Way ANOVA table is below:

The table above shows how to use the sums of squares and information from the group sizes to be able to calculate an F- statistic called F*.

Once we know F* we can compare it to the F-distribution to find a p-value. First we need a level of confidence, the degrees of freedom for the treatment and the degrees of freedom for the error to find the critical F. Thus,

If F* > F(0.95, degrees of freedom for treatment, degrees of freedom for error), we will reject the null hypothesis and conclude the alternative.

If F* < F(0.95, degrees of freedom for treatment, degrees of freedom for error), we will fail to reject the null hypothesis.

From here you interpret you results. If you rejected the null then you know that there is at least one group or treatment mean that differs from the others. If you fail to reject the null hypothesis then all the group means are equal. Just be sure to tell what that means in context of the problems.

AND THAT’S IT FOR ANOVA.

Usually this test is followed by what are called multiple comparisons, if one rejects the null. If the null hypothesis was not rejected then you are finished. Multiple comparisons will be the subject of future blogs.

FULL EXAMPLE

The following are generated data following sports example mentioned previously:

1) State Your Hypotheses

The null hypothesis is that all the teams have the same mean height:

The alternative is that at least one sports team differs in mean height:

2) Calculation of Sums of Square

SSTO (total sums of squares)

SSTO =(69-72.33)^2+(72-72.33)^2+ ...+ (79-72.33)^2+ (76-72.33)^2=270.7

SSTR (treatment sums of squares, sometimes called between)

SSTR = 10*(70-72.33)^2+(10*(72-72.33))^2+10*(75-72.33)^2=126.7

SSE (sums of squares of error, sometimes called within)

SSE = (69-70)^2+⋯+(69-70)^2+(74-72)^2+⋯+(67-72)^2+ (76-75)^2+⋯+ (76-75)^2=144

Notice -> SSTO = SSTR + SSE = 126.7 + 144 = 270.7

3) Create the ANOVA Table

4) Decision

If F* > F(0.95, 2, 27), we will reject the null hypothesis and conclude the alternative.

If F* < F(0.95, 2, 27), we will fail to reject the null hypothesis.

We need to find F(0.95, 2, 27). Using a table, or program, we find that F(0.95, 2, 27) = 3.354.

Because F* = 11.88 > 3.354 = F(0.95, 2, 27), we reject the null hypothesis that the group means are equal and go with the alternative.

5) Conclusion

Because we rejected the null hypothesis, we conclude the alternative: At least one team has a mean height different than another.

To figure out which one, we must perform multiple comparisons, the subject of a different blog.

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Mitosis!

by Shannen Ramey, Nicole Nesiba, andJon Zamora, CAPS Chemistry & Biology Tutors

Have you ever pondered over the biological topic of mitosis? If you have, then you are either taking a biology class or you have a strange thought process. In this blog, we are going to talk about the steps included in mitosis.

Let’s begin our discussion! What is mitosis you ask? Mitosis is the eukaryotic division of the nucleus, coupled with cytokinesis and the formation of two identical daughter cells. You are probably thinking this sounds really complex, but I’m here to break it down for you. It’s really not too bad! Just remember PMAT→ Prophase, metaphase, anaphase, and telophase.

Before mitosis begins the genetic material, or chromosomes, is replicated during a stage called interphase. From interphase, we enter the first step of PMAT-prophase. During prophase, chromosomes condense and the spindle apparatus begins to form. We have drawn the chromosomes in our illustration in a blue color surrounded by a light blue nuclear envelope, which is just beginning to break down. Notice, the cell now has several pairs of sister chromatids, which together form a chromosome. The yellow shows the spindle apparatus. 

Now we enter second step of PMAT- metaphase. By this phase, the nuclear envelope has completely broken down. The microtubules from the spindle apparatus have successfully caused the migration of chromosomes to the middle of the cell. Notice how the sister chromatids line up because this will be important for the next step of PMAT.

Welcome to anaphase! As you can see, the yellow spindle apparatus is pulling the sister chromatids apart from the previous step to opposite poles of the cell.

This brings us to our final step in PMAT, telophase. If you notice in the previous steps, the cell membrane has expanded as the chromosomes pull apart. Notice that the nuclear envelopes are beginning to reform and they each contain the separated chromosomes. Once the two independent nuclei have formed, mitosis is complete. At this point cells will continue to divide their cytoplasm via cytokinesis to form 2 daughter cells. Now we have completed MITOSIS and the daughter cells can now enter interphase in an ongoing process.

If we were to start this process again, do you remember at which stage we would start? If you thought interphase, then you are correct!

Well there you have it! We have made it through the entire process of mitosis. It wasn’t so bad, was it? It’s one of those processes that may look complicated at a first glance, but it’s not too bad once you take a closer look.

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Today in Science News: Gravitational Waves!

by Xavier Quintana, CAPS Assistant Coordinator for STEM

In 1921, Albert Einstein won the Nobel Prize in Physics for his work on the discovery of the photoelectric effect, specifically the description of how light interacts with matter. However, this discovery was only one of Einstein’s contributions to physics. In fact, when you think of Einstein you don’t think of light knocking electrons out of metal, you think of  the relation E=Mc2 and Special Relativity. He is most famous for his work on energy and matter.

Now, around 100 years after Einstein’s Theory of General Relativity was developed, physicists have proven Einstein’s largest contribution to our understanding of the universe.

What is General Relativity?

GR is the theory that our universe exists in the form of a sort-of fluid called space-time. Space-time consists of (as you might guess) a combination of our 3 dimensions of space (length, width, and height) as well as time. The geometry of spacetime is usually “flat”, except near anything with energy or mass. Energy and mass bend spacetime and cause gravity; the larger the mass, the larger the bend, the larger the gravity. Some people think that if you cut two holes in spacetime and connect them, you could teleport or time travel, these are called wormholes.

How do we know spacetime is real?

The earliest observations of evidence supporting general relativity came from a phenomenon called gravitational lensing. Astronomers observed that light was bent near very large objects, but the problem was that this bending couldn’t be explained by conventional physics. However, if you consider the fact that the light has to travel through bent spacetime, the lensing makes sense. Einstein also predicted a peculiar property of spacetime, much like a normal fluid, big changes in spacetime would cause ripples and waves to form.

How is a gravitational wave different from a water wave?

Water waves are an excellent analogy to gravitational waves, a small disturbance in the surface causes the water near it to move, and that causes a wave to move outward. However, a better analogy is sound. A constant, uniform sound source will push sound waves in all directions evenly by moving air molecules. If you make a big noise, like clap, there will be a very quick but intense sound wave. It is hard to hear a very weak, constant sound, but is easy to hear a clap even in a crowded room.

Discovery

Scientists from the Laser Interferometer Gravitational-Wave Observatory (LIGO) announced on February 11, 2016 that on September 14, 2015 that they heard a small clap in huge and crowded room. Two black holes collided somewhere in space and the energy of the collision caused an intense gravitational wave that was detected by two LIGO detectors. It took nearly 5 months to pour over the data and make sure that the tiny blip on the gravitational radar screen was the real deal. However, now that we know that gravitational waves are both real and detectable we can invent new ways to detect them.

Electromagnetic waves weren’t fully understood until the 1860s, 20 years later we learned how to make radio waves at will and 100 years later we had color TV in most homes in America. Will we be communicating via gravitational waves (like the film Interstellar) in 100 years? That is yet to be determined, but what is sure is that this discovery opens new doors to our understanding of the universe.

Xavier is CAPS’ Assistant Coordinator for STEM. He is a second-year Master’s student in the Department of Physics and Astronomy. He got his BS in Physics from New Mexico Institute of Mining and Technology in 2012 and worked in Washington, DC before coming back to Albuquerque. He enjoys skiing and international politics, and finds ways to utilize physics in both.

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The Physics of Black Holes

by Jordan Van Every, CAPS Math and Physics Tutor

What is Black Hole?

Black holes are one of the most popularized ideas of contemporary physics, but are misunderstood by the general public. Colloquially, they’re considered objects shrouded in mystery that have the ability to absorb everything in their wake. While this isn’t totally incorrect, astrophysicists actually understand some of the mathematics behind black holes and know how one might potentially form.

 An artist’s rendition of black hole.

Check out this video to learn more about black holes!

So Why are Black Holes so Strange?

For one, a black hole can’t be observed directly! An observation in astrophysics generally means that light has bounced off the object you’re looking at and been received by your eye, or some other instrument like a telescope, but black holes absorb all the light that comes within a certain distance to them. So how do we even know they’re there in the first place?

Even stranger, light can’t escape a black hole? Light, or now generally thought of as ‘photons’, is the fastest moving object in the universe to our knowledge, and travels at a speed:

So what’s the problem? You’d think that a black hole wouldn’t care how fast something was going once it reached a certain size right? The problem here is…..  photons don’t have any mass! The reason that things can’t escape black holes is because of gravitation, attraction between two bodies with mass, so how does this happen?

Universal Gravitation:

Back in the early years of physics, Issac Newton hypothesized a few things from his observations of bodies in nature, ones that you’ve most likely heard before. Among his three laws of motion, Newton hypothesized this:

“Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.”

Or in the language of mathematics:

In casual terms this means: The force between two objects with mass depends on how big they are, and how far their separation is.

So putting a photon mass of 0 into this equation makes the whole thing zero. No force, means no acceleration, or ‘change in motion’ by Newton’s second law. So we would expect photons to whiz right by black holes, or reflect off of them, making them observable. So something is wrong here with the way we think about black holes. There are only two possible explanations to remedy this issue.

1.    Black holes simply don’t exist! Because if we can observe them with light, then it isn’t a black hole anymore.

2.    Maybe photons can be affected by gravity for some other reason? Maybe there is a problem with universal gravitation?

Things We Know:

1. Light bends in a gravitational field! This was predicted by Einstein’s General Theory of Relativity, and later confirmed by observation of the ‘Twin Quasar’.

“an intervening mass concentration between Earth and the quasar bends light so that two images of the quasar appear in the sky. This is known as gravitational lensing, and is a consequence of Einsteinian warped space-time.”

2. Theoretically, black holes can exist, and Astrophysicists think they form from the death of ‘Neutron Stars’. The idea is, that when a Neutron star collapses, it reduces to a super dense ‘singularity’, meaning that you condense multiple solar masses (our sun’s mass) into a small space, and the gravitation produced is so large that a black hole forms.

3. There is a point of no return! It’s true, there is a theorized distance from the center of black hole that is the closest any object in the universe can get before getting pulled in with no chance of escape. This is called the ‘Schwarzschild Radius’, and it is associated with the ‘photon energy’ instead of the photon mass. (A problem that we discussed earlier)

This is also known as the ‘event horizon’ or ‘point of no return’.

How Can we Know They Exist?

This is still an issue of observation to be handled with black holes. Since we can’t actually look at them, we have to detect them another way. A potentially convincing example of a black hole is near the blue supergiant Cygnus X-1. This star is about 25 times the mass of the sun, but it seems to be orbiting a non-luminous x-ray source, which is hypothesized to be a neutron star or a black hole. Find out more about Cygnus X-1 on Hyper Physics.

Black holes are a hot topic of interest in modern astrophysics, and there is certainly much more to know about these beasts of the cosmos.

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How to Win at Angry Birds

Source: Mashable

Valuable vectors:

When dealing with the motion of an object in 2 dimensions, also called projectile motion, we take advantage of one fascinating feature of vectors- components! Recall that a vector has direction and magnitude; this means that the vector has a length (magnitude) and is at an angle with “something else” (direction). We normally use the ground as that “something else” to measure angle. We can separate a vector into two new vectors (called components) by using some ingenious geometry:

vector into two new vectors (called components) by using some ingenious geometry:

Once we have our components, we can significantly simplify our situation. The vectors we care about in this case are position, velocity, and acceleration.

Distinct Dimensions:

By separating the vectors into components, we can consider motion along those vectors separately. The motion along x alone won’t depend on the motion along y alone, and vice versa. Once we have our components, we can use the usual 1-D kinematic equations for the two components separately:

In the x-direction:

and in the y-direction:

It is very important to keep the x and y components separate while you are calculating the motion. If your final answer must be in terms of a net vector, you can use the same trigonometry above to solve for the magnitudes and directions of your final vectors.

Any questions? Please visit http://caps.unm.edu/hours/subject.php?name=physics for information on Physics Drop-In and Online hours at CAPS!

Useful links:

Vectors

1-D and 2-D motion

Simulation

Xavier is a first-year Master’s student in the Department of Physics and Astronomy (PANDA). He got his BS in Physics from New Mexico Institute of Mining and Technology in 2012 and worked in Washington, DC before coming back to Albuquerque. He enjoys skiing and international politics, and finds ways to utilize physics in both.