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Why Relations between Variables are Important

Why Relations between Variables are Important

Generally speaking, the ultimate goal of every research or scientific analysis is to find relations between variables. The philosophy of science teaches us that there is no other way of representing "meaning" except in terms of relations between some quantities or qualities; either way involves relations between variables. Thus, the advancement of science must always involve finding new relations between variables. Correlational research involves measuring such relations in the most straightforward manner. However, experimental research is not any different in this respect. For example, the above mentioned experiment comparing WCC in males and females can be described as looking for a correlation between two variables: Gender and WCC. Statistics does nothing else but help us evaluate relations between variables. Actually, all of the hundreds of procedures that are described in this online textbook can be interpreted in terms of evaluating various kinds of inter-variable relations.

How to Measure the Magnitude (Strength) of Relations between Variables

How to Measure the Magnitude (Strength) of Relations between Variables

There are very many measures of the magnitude of relationships between variables that have been developed by statisticians; the choice of a specific measure in given circumstances depends on the number of variables involved, measurement scales used, nature of the relations, etc. Almost all of them, however, follow one general principle: they attempt to somehow evaluate the observed relation by comparing it to the "maximum imaginable relation" between those specific variables.

Technically speaking, a common way to perform such evaluations is to look at how differentiated the values are of the variables, and then calculate what part of this "overall available differentiation" is accounted for by instances when that differentiation is "common" in the two (or more) variables in question. Speaking less technically, we compare "what is common in those variables" to "what potentially could have been common if the variables were perfectly related."

Let's consider a simple illustration. Let's say that in our sample, the average index of WCC is 100 in males and 102 in females. Thus, we could say that on average, the deviation of each individual score from the grand mean (101) contains a component due to the gender of the subject; the size of this component is 1. That value, in a sense, represents some measure of relation between Gender and WCC. However, this value is a very poor measure because it does not tell us how relatively large this component is given the "overall differentiation" of WCC scores. Consider two extreme possibilities:

1. If all WCC scores of males were equal exactly to 100 and those of females equal to 102, then all deviations from the grand mean in our sample would be entirely accounted for by gender. We would say that in our sample, Gender is perfectly correlated with WCC, that is, 100% of the observed differences between subjects regarding their WCC is accounted for by their gender.
    
2. If WCC scores were in the range of 0-1000, the same difference (of 2) between the average WCC of males and females found in the study would account for such a small part of the overall differentiation of scores that most likely it would be considered negligible. For example, one more subject taken into account could change, or even reverse the direction of the difference. Therefore, every good measure of relations between variables must take into account the overall differentiation of individual scores in the sample and evaluate the relation in terms of (relatively) how much of this differentiation is accounted for by the relation in question.

Common "General Format" of Most Statistical Tests

Common "General Format" of Most Statistical Tests

Because the ultimate goal of most statistical tests is to evaluate relations between variables, most statistical tests follow the general format that was explained in the previous paragraph. Technically speaking, they represent a ratio of some measure of the differentiation common in the variables in question to the overall differentiation of those variables. For example, they represent a ratio of the part of the overall differentiation of the WCC scores that can be accounted for by gender to the overall differentiation of the WCC scores. This ratio is usually called a ratio of explained variation to total variation. In statistics, the term explained variation does not necessarily imply that we "conceptually understand" it. It is used only to denote the common variation in the variables in question, that is, the part of variation in one variable that is "explained" by the specific values of the other variable, and vice versa.

How the "Level of Statistical Significance" is Calculated

How the "Level of Statistical Significance" is Calculated

Let's assume that we have already calculated a measure of a relation between two variables (as explained above). The next question is "how significant is this relation?" For example, is 40% of the explained variance between the two variables enough to consider the relation significant? The answer is "it depends."

Specifically, the significance depends mostly on the sample size. As explained before, in very large samples, even very small relations between variables will be significant, whereas in very small samples even very large relations cannot be considered reliable (significant). Thus, in order to determine the level of statistical significance, we need a function that represents the relationship between "magnitude" and "significance" of relations between two variables, depending on the sample size. The function we need would tell us exactly "how likely it is to obtain a relation of a given magnitude (or larger) from a sample of a given size, assuming that there is no such relation between those variables in the population." In other words, that function would give us the significance (p) level, and it would tell us the probability of error involved in rejecting the idea that the relation in question does not exist in the population. This "alternative" hypothesis (that there is no relation in the population) is usually called the null hypothesis. It would be ideal if the probability function was linear and, for example, only had different slopes for different sample sizes. Unfortunately, the function is more complex and is not always exactly the same; however, in most cases we know its shape and can use it to determine the significance levels for our findings in samples of a particular size. Most of these functions are related to a general type of function, which is called normal.

Why the "Normal Distribution" is Important

The "normal distribution" is important because in most cases, it well approximates the function that was introduced in the previous paragraph (for a detailed illustration, see Are All Test Statistics Normally Distributed?). The distribution of many test statistics is normal or follows some form that can be derived from the normal distribution. In this sense, philosophically speaking, the normal distribution represents one of the empirically verified elementary "truths about the general nature of reality," and its status can be compared to the one of fundamental laws of natural sciences. The exact shape of the normal distribution (the characteristic "bell curve") is defined by a function that has only two parameters: mean and standard deviation.

A characteristic property of the normal distribution is that 68% of all of its observations fall within a range of ±1 standard deviation from the mean, and a range of ±2 standard deviations includes 95% of the scores. In other words, in a normal distribution, observations that have a standardized value of less than -2 or more than +2 have a relative frequency of 5% or less. (Standardized value means that a value is expressed in terms of its difference from the mean, divided by the standard deviation.) If you have access to STATISTICA, you can explore the exact values of probability associated with different values in the normal distribution using the interactive Probability Calculator tool; for example, if you enter the Z value (i.e., standardized value) of 4, the associated probability computed bySTATISTICA will be less than .0001, because in the normal distribution almost all observations (i.e., more than 99.99%) fall within the range of ±4 standard deviations. The animation below shows the tail area associated with other Z values.

Illustration of How the Normal Distribution is Used in Statistical Reasoning (Induction)

Recall the example discussed above, where pairs of samples of males and females were drawn from a population in which the average value of WCC in males and females was exactly the same. Although the most likely outcome of such experiments (one pair of samples per experiment) was that the difference between the average WCC in males and females in each pair is close to zero, from time to time, a pair of samples will be drawn where the difference between males and females is quite different from 0. How often does it happen? If the sample size is large enough, the results of such replications are "normally distributed" (this important principle is explained and illustrated in the next paragraph) and, thus, knowing the shape of the normal curve, we can precisely calculate the probability of obtaining "by chance" outcomes representing various levels of deviation from the hypothetical population mean of 0. If such a calculated probability is so low that it meets the previously accepted criterion of statistical significance, then we have only one choice: conclude that our result gives a better approximation of what is going on in the population than the "null hypothesis" (remember that the null hypothesis was considered only for "technical reasons" as a benchmark against which our empirical result was evaluated). Note that this entire reasoning is based on the assumption that the shape of the distribution of those "replications" (technically, the "sampling distribution") is normal. This assumption is discussed in the next paragraph.

Are All Test Statistics Normally Distributed?

Are All Test Statistics Normally Distributed?

Not all, but most of them are either based on the normal distribution directly or on distributions that are related to and can be derived from normal, such as t, F, or Chi-square. Typically, these tests require that the variables analyzed are themselves normally distributed in the population, that is, they meet the so-called "normality assumption." Many observed variables actually are normally distributed, which is another reason why the normal distribution represents a "general feature" of empirical reality. The problem may occur when we try to use a normal distribution-based test to analyze data from variables that are themselves not normally distributed (see tests of normality in Nonparametrics or ANOVA/MANOVA). In such cases, we have two general choices. First, we can use some alternative "nonparametric" test (or so-called "distribution-free test" see, Nonparametrics); but this is often inconvenient because such tests are typically less powerful and less flexible in terms of types of conclusions that they can provide. Alternatively, in many cases we can still use the normal distribution-based test if we only make sure that the size of our samples is large enough. The latter option is based on an extremely important principle that is largely responsible for the popularity of tests that are based on the normal function. Namely, as the sample size increases, the shape of the sampling distribution (i.e., distribution of a statistic from the sample; this term was first used by Fisher, 1928a) approaches normal shape, even if the distribution of the variable in question is not normal. This principle is illustrated in the following animation showing a series of sampling distributions (created with gradually increasing sample sizes of: 2, 5, 10, 15, and 30) using a variable that is clearly non-normal in the population, that is, the distribution of its values is clearly skewed.

 However, as the sample size (of samples used to create the sampling distribution of the mean) increases, the shape of the sampling distribution becomes normal. Note that for n=30, the shape of that distribution is "almost" perfectly normal (see the close match of the fit). This principle is called the central limit theorem (this term was first used by Pólya, 1920; German, "Zentraler Grenzwertsatz").

relationships between variables statistics

How Do We Know the Consequences of Violating the Normality Assumption?

Although many of the statements made in the preceding paragraphs can be proven mathematically, some of them do not have theoretical proof and can be demonstrated only empirically, via so-called Monte-Carlo experiments. In these experiments, large numbers of samples are generated by a computer following predesigned specifications, and the results from such samples are analyzed using a variety of tests. This way we can empirically evaluate the type and magnitude of errors or biases to which we are exposed when certain theoretical assumptions of the tests we are using are not met by our data. Specifically, Monte-Carlo studies were used extensively with normal distribution-based tests to determine how sensitive they are to violations of the assumption of normal distribution of the analyzed variables in the population. The general conclusion from these studies is that the consequences of such violations are less severe than previously thought. Although these conclusions should not entirely discourage anyone from being concerned about the normality assumption, they have increased the overall popularity of the distribution-dependent statistical tests in all areas of research.

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Relational model

The purpose of the relational model is to provide a declarative method for specifying data and queries: users directly state what information the database contains and what information they want from it, and let the database management system software take care of describing data structures for storing the data and retrieval procedures for answering queries.

IBM's original implementation of Codd's ideas was System R. There have been several commercial and open source products based on Codd's ideas, including IBM's DB2,Oracle Database, Microsoft SQL Server, PostgreSQL, MySQL, and many others. Most of these use the SQL data definition and query language. A table in an SQL database schema corresponds to a predicate variable; the contents of a table to a relation; key constraints, other constraints, and SQL queries correspond to predicates. However, SQL databases, including DB2, deviate from the relational model in many details; Codd fiercely argued against deviations that compromise the original principles

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Reliability & Validity

We often think of reliability and validity as separate ideas but, in fact, they're related to each other. Here, I want to show you two ways you can think about their relationship.

One of my favorite metaphors for the relationship between reliability is that of the target. Think of the center of the target as the concept that you are trying to measure. Imagine that for each person you are measuring, you are taking a shot at the target. If you measure the concept perfectly for a person, you are hitting the center of the target. If you don't, you are missing the center. The more you are off for that person, the further you are from the center.

The figure above shows four possible situations. In the first one, you are hitting the target consistently, but you are missing the center of the target. That is, you are consistently and systematically measuring the wrong value for all respondents. This measure is reliable, but no valid (that is, it's consistent but wrong). The second, shows hits that are randomly spread across the target. You seldom hit the center of the target but, on average, you are getting the right answer for the group (but not very well for individuals). In this case, you get a valid group estimate, but you are inconsistent. Here, you can clearly see that reliability is directly related to the variability of your measure. The third scenario shows a case where your hits are spread across the target and you are consistently missing the center. Your measure in this case is neither reliable nor valid. Finally, we see the "Robin Hood" scenario -- you consistently hit the center of the target. Your measure is both reliable and valid (I bet you never thought of Robin Hood in those terms before).

Another way we can think about the relationship between reliability and validity is shown in the figure below. Here, we set up a 2x2 table. The columns of the table indicate whether you are trying to measure the same or different concepts. The rows show whether you are using the same or different methods of measurement. Imagine that we have two concepts we would like to measure, student verbal and math ability. Furthermore, imagine that we can measure each of these in two ways. First, we can use a written, paper-and-pencil exam (very much like the SAT or GRE exams). Second, we can ask the student's classroom teacher to give us a rating of the student's ability based on their own classroom observation.

The first cell on the upper left shows the comparison of the verbal written test score with the verbal written test score. But how can we compare the same measure with itself? We could do this by estimating the reliability of the written test through a test-retest correlation, parallel forms, or an internal consistency measure (See Types of Reliability). What we are estimating in this cell is the reliability of the measure.

The cell on the lower left shows a comparison of the verbal written measure with the verbal teacher observation rating. Because we are trying to measure the same concept, we are looking at convergent validity (See Measurement Validity Types).

The cell on the upper right shows the comparison of the verbal written exam with the math written exam. Here, we are comparing two different concepts (verbal versus math) and so we would expect the relationship to be lower than a comparison of the same concept with itself (e.g., verbal versus verbal or math versus math). Thus, we are trying to discriminate between two concepts and we would consider this discriminant validity.

Finally, we have the cell on the lower right. Here, we are comparing the verbal written exam with the math teacher observation rating. Like the cell on the upper right, we are also trying to compare two different concepts (verbal versus math) and so this is a discriminant validity estimate. But here, we are also trying to compare two different methods of measurement (written exam versus teacher observation rating). So, we'll call this very discriminant to indicate that we would expect the relationship in this cell to be even lower than in the one above it.

The four cells incorporate the different values that we examine in the multitrait-multimethod approach to estimating construct validity.

When we look at reliability and validity in this way, we see that, rather than being distinct, they actually form a continuum. On one end is the situation where the concepts and methods of measurement are the same (reliability) and on the other is the situation where concepts and methods of measurement are different (verydiscriminant validity).

How to be more Reliable in Your Relationships

How to be more Reliable in Your Relationships

In relationships you want to be reliable. Reliability of our partner is a primary concern for all of us. We want them to be reliable in many ways. We want them to show up when they say they will. We want them to no cheat. We want them to maintain their job, etc.

So, how can you be more reliable in your relationships? 

You can be more reliable by only making promises you intend to keep. Do not tell your partner something that is untrue, even if it is only half untrue.

You can be more reliable by following through with your feelings. If you want to be reliable and you say you love someone, you have to act like you love the person. Words and actions being consistent is part of being reliable. If you say you love your partner and then you abuse them, you are not reliable.

You can be more reliable by using your cell phone What? If you are going to be late, pick up your cell phone and call your partner and let them know, do not let them worry. If you can't do something they asked of you, tell them. Do not just wait until they ask why you did not do it.

You can be more reliable by making your actions consistent. So, if you show compassion to your partner when they get in a bad situation, but blame the homeless for their situation you are not being consistent. Eventually something is going to change. So, make all of your actions be what you want to have with your partner.

You being reliable is not enough, you want your partner to be reliable as well. How can you know if your partner is reliable? Try the following quiz:

Quiz your partner's basic values. Your values are going to say a lot about how reliable they are. Is your partner an opportunist? Do they take advantage of the best deal, or do they stick with their word on things? Let's look at a small example. Let's say you are shopping and your partner wants to buy a new suit. They try it on, it fits, they like it, they tell the sales person they will return and buy it when they are finished shopping. While you are shopping you find the same suit for less. So, does he jump on the opportunity and buy the suit that costs less, or does he keep his word? This may look and sound something very trivial, but character is always seen in the smallest instances. If you want your partner to be reliable look at their reliability in other instances.

Quiz your partner's ability to be compassionate. Does your partner see a homeless person and say they should get a job, or do they feel bad that they have fallen on hard times? When they see someone in need what is their reaction? Do they show compassion or do they place blame on the person? You may want to recognize that the way they act could be directed toward you some day. This is important to recognize because you want them to be reliable about staying consistent with their treatment of you and their compassion for you.

Next, quiz all of your partner's character traits- from love, to past relationships, to compassion on the elderly, understanding of various emotions, intelligence, values etc. and find out if your partner is reliable. To find out if your partner is reliable you will want to look at many things, and how they treat people is one of those things. If you ignore the small signs it will be at our own cost.

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Relationship Quotes

Health is the greatest gift, contentment the greatest wealth, faithfulness the best relationship.
Buddha

Spiritual relationship is far more precious than physical. Physical relationship divorced from spiritual is body without soul.
Mohandas Gandhi 

If you haven't found it yet, keep looking. Don't settle. As with all matters of the heart, you'll know when you find it. And, like any great relationship, it just gets better and better as the years roll on.
Steve Jobs 

When you make the sacrifice in marriage, you're sacrificing not to each other but to unity in a relationship.
Joseph Campbell

It is of practical value to learn to like yourself. Since you must spend so much time with yourself you might as well get so out of the relationship.
Norman Vincent Peale

There is no more lovely, friendly and charming relationship, communion or company than a good marriage.
Martin Luther

Sister is probably the most competitive relationship within the family, but once the sisters are grown, it becomes the strongest relationship.
Margaret Mead

A person isn't who they are during the last conversation you had with them - they're who they've been throughout your whole relationship.
Rainer Maria Rilke

So then, the relationship of self to other is the complete realization that loving yourself is impossible without loving everything defined as other than yourself.
Alan Watts

If the relationship of father to son could really be reduced to biology, the whole earth would blaze with the glory of fathers and sons.
James A. Baldwin
As soon as I got out there I felt a strange relationship with the pitcher's mound. It was as if I'd been born out there. Pitching just felt like the most natural thing in the world. Striking out batters was easy.
Babe Ruth

My relationship to power and authority is that I'm all for it. People need somebody to watch over them. Ninety-five percent of the people in the world need to be told what to do and how to behave.
Arnold Schwarzenegger

Whenever you're in conflict with someone, there is one factor that can make the difference between damaging your relationship and deepening it. That factor is attitude.
William James

If one could be friendly with women, what a pleasure - the relationship so secret and private compared with relations with men. Why not write about it truthfully?
Virginia Woolf

The first test of a truly great man is his humility. By humility I don't mean doubt of his powers or hesitation in speaking his opinion, but merely an understanding of the relationship of what he can say and what he can do.
John Ruskin

The relationship between a manufacturer and his advertising agency is almost as intimate as the relationship between a patient and his doctor. Make sure that you can life happily with your prospective client before you accept his account.
David Ogilvy

The relationship to one's fellow man is the relationship of prayer, the relationship to oneself is the relationship of striving; it is from prayer that one draws the strength for one's striving.
Franz Kafka

Never assume that the guy understands that you and he have a relationship.
Dave Barry

We have a strange and wonderful relationship - he's strange and I'm wonderful.
Mike Ditka

Courage means to keep working a relationship, to continue seeking solutions to difficult problems, and to stay focused during stressful periods. 

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Brother-in-Law, Sister-in-Law

These are the only really tricky in-law terms. “Brother-in-law” and “sister-in-law” each have two or three meanings. All authorities agree on the first two meanings, but there is some controversy about the third (and I personally don’t accept it).

My sister-in-law could be:

1. the sister of my spouse, or
2. the wife of my brother, or
3. the wife of my spouse’s brother. (This meaning is accepted by The American Heritage Dictionary of the English Language (Third Edition, 1992), but not by all authorities.)

Similarly, my brother-in-law could be

1. the brother of my spouse, or
2. the husband of my sister, or
3. the husband of my spouse’s sister. (This meaning is accepted by The American Heritage Dictionary of the English Language (Third Edition, 1992), but not by all authorities.)

Consider the following example: Al marries Betty; Betty has a sister Bonnie, who marries Calvin.

         Harry = Sally                                |                                      +                             --------------------                   |                  |             Al = Betty            Bonnie = Calvin 

The siblings-in-law:

• Al is Bonnie’s brother-in-law (definition 2), and Bonnie is Al’s sister-in-law (definition 1);
• Betty is Calvin’s sister-in-law (definition 1), and Calvin is Betty’s brother-in-law (definition 2).

So much is agreed. The question is, are Al and Calvin brothers-in-law (definition 3)? Someone once wrote to Ann Landers, the advice columnist, describing this situation. Ann replied: “You are no relation; you are just two men who married sisters.” Though I agree with Ann on this one, I admit that it’s awkward for Al to refer to Calvin as “my wife’s brother-in-law” or “my sister-in-law’s husband”. Probably that’s why Al might refer to Calvin as “my brother-in-law”.

Aunt, Uncle, Niece, Nephew

There are standard words for collateral relationships, where neither person is directly descended from the other. B0 and C0 are brother and sister, or more generically siblings. Going down one generation, on one side only, we have four common relationships:

• B0 is C1‘s uncle. Most English speakers use “uncle” for any of four relationships: father’s brother, mother’s brother, father’s sister’s husband, or mother’s sister’s husband.
• C0 is B1‘s aunt. Again, “aunt” in English could mean your father’s sister, mother’s sister, father’s brother’s wife, or mother’s brother’s wife. Most people would say Z0 is also C1‘s aunt.
• C1 is the niece or nephew of B0; most people would say C1 is also the niece or nephew of Z0.

Children of your aunt or uncle are your first cousins. More generally, B1 and C1, and all their descendants, are cousins to each other. A separate section below details all the words used to describe cousin relationships.