What does "independent observations" mean?

Question

I'm trying to understand what the assumption of independent observations means. Some definitions are:

1. "Two events are independent if and only if $P(a \cap b) = P(a) * P(b)$." (Statistical Terms Dictionary)
2. "the occurrence of one event doesn't change the probability for another" (Wikipedia).
3. "sampling of one observation does not affect the choice of the second observation" (David M. Lane).

An example of dependent observations that's often given is students nested within teachers as below. Let's assume that teachers influence students but students don't influence one another.

So how are these definitions violated for these data? Sampling [grade = 7] for [student = 1] does not affect the probability distribution for the grade that will be sampled next. (Or does it? And if so, then what does observation 1 predict regarding the next observation?)

Why would the observations be independent if I had measured gender instead of teacher_id? Don't they affect the observations in the same way?

teacher_id   student_id   grade
         1            1       7
         1            2       7
         1            3       6
         2            4       8
         2            5       8
         2            6       9

Answer 1

In probability theory, statistical independence (which is not the same as causal independence) is defined as your property (3), but (1) follows as a consequence$\dagger$. The events $\mathcal{A}$ and $\mathcal{B}$ are said to be statistically independent if and only if:

$$\mathbb{P}(\mathcal{A} \cap \mathcal{B}) = \mathbb{P}(\mathcal{A}) \cdot \mathbb{P}(\mathcal{B}) .$$

If $\mathbb{P}(\mathcal{B}) > 0$ then if follows that:

$$\mathbb{P}(\mathcal{A} |\mathcal{B}) = \frac{\mathbb{P}(\mathcal{A} \cap \mathcal{B})}{\mathbb{P}(\mathcal{B})} = \frac{\mathbb{P}(\mathcal{A}) \cdot \mathbb{P}(\mathcal{B})}{\mathbb{P}(\mathcal{B})} = \mathbb{P}(\mathcal{A}) .$$

This means that statistical independence implies that the occurrence of one event does not affect the probability of the other. Another way of saying this is that the occurrence of one event should not change your beliefs about the other. The concept of statistical independence is generally extended from events to random variables in a way that allows analogous statements to be made for random variables, including continuous random variables (which have zero probability of any particular outcome). Treatment of independence for random variables basically involves the same definitions applied to distribution functions.

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It is crucial to understand that independence is a very strong property - if events are statistically independent then (by definition) we cannot learn about one from observing the other. For this reason, statistical models generally involve assumptions of conditional independence, given some underlying distribution or parameters. The exact conceptual framework depends on whether one is using Bayesian methods or classical methods. The former involves explicit dependence between observable values, while the latter involves a (complicated and subtle) implicit form of dependence. Understanding this issue properly requires a bit of understanding of classical versus Bayesian statistics.

Statistical models will often say they use an assumption that sequences of random variables are "independent and identically distributed (IID)". For example, you might have an observable sequence $X_1, X_2, X_3, ... \sim \text{IID N} (\mu, \sigma^2)$, which means that each observable random variable $X_i$ is normally distributed with mean $\mu$ and standard deviation $\sigma$. Each of the random variables in the sequence is "independent" of the others in the sense that its outcome does not change the stated distribution of the other values. In this kind of model we use the observed values of the sequence to estimate the parameters in the model, and we can then in turn predict unobserved values of the sequence. This necessarily involves using some observed values to learn about others.

Bayesian statistics: Everything is conceptually simple. Assume that $X_1, X_2, X_3, ... $ are conditionally IID given the parameters $\mu$ and $\sigma$, and treat those unknown parameters as random variables. Given any non-degenerate prior distribution for these parameters, the values in the observable sequence are (unconditionally) dependent, generally with positive correlation. Hence, it makes perfect sense that we use observed outcomes to predict later unobserved outcomes - they are conditionally independent, but unconditionally dependent.

Classical statistics: This is quite complicated and subtle. Assume that $X_1, X_2, X_3, ... $ are IID given the parameters $\mu$ and $\sigma$, but treat those parameters as "unknown constants". Since the parameters are treated as constants, there is no clear difference between conditional and unconditional independence in this case. Nevertheless, we still use the observed values to estimate the parameters and make predictions of the unobserved values. Hence, we use the observed outcomes to predict later unobserved outcomes even though they are notionally "independent" of each other. This apparent incongruity is discussed in detail in O'Neill, B. (2009) Exchangeability, Correlation and Bayes' Effect. International Statistical Review 77(2), pp. 241 - 250.

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Applying this to your student grades data, you would probably model something like this by assuming that grade is conditionally independent given teacher_id. You would use the data to make inferences about the grading distribution for each teacher (which would not be assumed to be the same) and this would allow you to make predictions about the unknown grade of another student. Because the grade variable is used in the inference, it will affect your predictions of any unknown grade variable for another student. Replacing teacher_id with gender does not change this; in either case you have a variable that you might use as a predictor of grade.

If you use Bayesian method you will have an explicit assumption of conditional independence and a prior distribution for the teachers' grade distributions, and this leads to unconditional (predictive) dependence of grades, allowing you to rationally use one grade in your prediction of another. If you are using classical statistics you will have an assumption of independence (based on parameters that are "unknown constants") and you will use classical statistical prediction methods that allow you to use one grade to predict another.

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$\dagger$ There are some foundational presentations of probability theory that define independence via the conditional probability statement and then give the joint probability statement as a consequence. This is less common.

Answer 2

The definitions of statistical independence that you give in your post are all essentially correct, but they don't get to the heart of the independence assumption in a statistical model. To understand what we mean by the assumption of independent observations in a statistical model, it will be helpful to revisit what a statistical model is on a conceptual level.

Statistical models as approximations to "nature's dice"

Let's use a familiar example: we collect a random sample of adult humans (from a well-defined population--say, all adult humans on earth) and we measure their heights. We wish to estimate the population mean height of adult humans. To do this, we construct a simple statistical model by assuming that people's heights arise from a normal distribution.

Our model will be a good one if a normal distribution provides a good approximation to how nature "picks" heights for people. That is, if we simulate data under our normal model, does the resulting dataset closely resemble (in a statistical sense) what we observe in nature? In the context of our model, does our random-number generator provide a good simulation of the complicated stochastic process that nature uses to determine the heights of randomly selected human adults ("nature's dice")?

The independence assumption in a simple modeling context

When we assumed that we could approximate "nature's dice" by drawing random numbers from a normal distribution, we didn't mean that we would draw a single number from the normal distribution, and then assign that height to everybody. We meant that we would independently draw numbers for everybody from the same normal distribution. This is our independence assumption.

Imagine now that our sample of adults wasn't a random sample, but instead came from a handful of families. Tallness runs in some families, and shortness runs in others. We've already said that we're willing to assume that the heights of all adults come from one normal distribution. But sampling from the normal distribution wouldn't provide a dataset that looks much like our sample (our sample would show "clumps" of points, some short, others tall--each clump is a family). The heights of people in our sample are not independent draws from the overall normal distribution.

The independence assumption in a more complicated modeling context

But not all is lost! We might be able to write down a better model for our sample--one that preserves the independence of the heights. For example, we could write down a linear model where heights arise from a normal distribution with a mean that depends on what family the subject belongs to. In this context, the normal distribution describes the residual variation, AFTER we account for the influence of family. And independent samples from a normal distribution might be a good model for this residual variation.

Overall here, what we have done is to write down a more sophisticated model of how we expect nature's dice to behave in the context of our study. By writing down a good model, we might still be justified in assuming that that the random part of the model (i.e. the random variation around the family means) is independently sampled for each member of the population.

The (conditional) independence assumption in a general modeling context

In general, statistical models work by assuming that data arises from some probability distribution. The parameters of that distribution (like the mean of the normal distribution in the example above) might depend on covariates (like family in the example above). But of course endless variations are possible. The distribution might not be normal, the parameter that depends on covariates might not be the mean, the form of the dependence might not be linear, etc. ALL of these models rely on the assumption that they provide a reasonably good approximation to how nature's dice behave (again, that data simulated under the model will look statistically similar to actual data obtained by nature).

When we simulate data under the model, the final step will always be to draw a random number according to some modeled probability distribution. These are the draws that we assume to be independent of one another. The actual data that we get out might not look independent, because covariates or other features of the model might tell us to use different probability distributions for different draws (or sets of draws). But all of this information must be built into the model itself. We are not allowed to let the random final number draw depend on what values we drew for other data points. Thus, the events that need to be independent are the rolls of "nature's dice" in the context of our model.

It is useful to refer to this situation as conditional independence, which means that the data points are independent of one another given (i.e. conditioned on) the covariates. In our height example, we assume my height and my brother's height conditioned on my family are independent of one another, and are also independent of your height and your sister's height conditioned on your family. Once we know somebody's family, we know what normal distribution to draw from to simulate their height, and the draws for different individuals are independent regardless of their family (even though our choice of what normal distribution to draw from depends on family). It's also possible that even after dealing with the family structure of our data, we still don't achieve good conditional independence (maybe it's also important to model gender, for example).

Ultimately, whether it makes sense to assume conditional independence of observations is a decision that must be undertaken in the context of a particular model. This is why, for example, in linear regression, we don't check that the data come from a normal distribution, but we do check that the RESIDUALS come from a normal distribution (and from the SAME normal distribution across the full range of the data). The linear regression assumes that, after accounting for the influence of covariates (the regression line), the data are independently sampled from a normal distribution, according to the strict definition of independence in the original post.

In the context of your example

"Teacher" in your data might be like "family" in the height example.

A final spin on it

Lots of familiar models assume that the residuals arise from a normal distribution. Imagine I gave you some data that very clearly were NOT normal. Maybe the're strongly skewed, or maybe they're bimodal. And I told you "these data come from a normal distribution."

"No way," you say, "It's obvious that those aren' normal!"

"Who said anything about the data being normal?" I say. "I only said that they come from a normal distribution."

"One in the same!" you say. "We know that a histogram of reasonably large sample from a normal distribution will tend to look approximately normal!"

"But," I say, "I never said the data were independently sampled from the normal distribution. The DO come from a normal distribution, but they aren't independent draws."

The assumption of (conditional) independence in statistical modeling is there to prevent smart-alecks like me from ignoring the distribution of the residuals and mis-applying the model.

Two final notes

1) The term "nature's dice" is not mine originally, but despite consulting a couple of references I can't figure out where I got it in this context.

2) Some statistical models (e.g. autoregressive models) do not require independence of observations in quite this way. In particular, they allow the sampling distribution for a given observation to depend not only fixed covariates, but also on the data that came before it.

Answer 3

Let $\mathbb x=(X_1,...,X_j,...,X_k)$ by a $k-$dimensional random vector, i.e. a fixed-position collection of random variables (measurable real functions).

Consider many such vectors, say $n$, and index these vectors by $i=1,...,n$, so, say

$$\mathbb x_i=(X_{1i},...,X_{ji},...,X_{ki})$$ and regard them as a collection called "the sample", $S=(\mathbb x_1,...,\mathbb x_i,...,\mathbb x_n)$. Then we call each $k-$ dimensional vector an "observation" (although it really becomes one only once we measure and record the realizations of the random variables involved ).

Let's first treat the case where either a probability mass function (PMF) or a probability density function (PDF) exists, and also, joint such functions. Denote by $f_i(\mathbb x_i),\;i=1,...,n$ the joint PMF or joint PDF of each random vector, and $f(\mathbb x_1,...,\mathbb x_i,...,\mathbb x_n)$ the joint PMF or joint PDF of all these vectors together.

Then, the sample $S$ is called an "independent sample", if the following mathematical equality holds:

$$f(\mathbb x_1,...,\mathbb x_i,...,\mathbb x_n) = \prod_{i=1}^{n}f_i(\mathbb x_i),\;\;\; \forall (\mathbb x_1,...,\mathbb x_i,...,\mathbb x_n) \in D_S$$

where $D_S$ is the joint domain created by the $n$ random vectors/observations.

This means that the "observations" are "jointly independent", (in the statistical sense, or "independent in probability" as was the old saying that is still seen today sometimes). The habit is to simply call them "independent observations".

Note that the statistical independence property here is over the index $i$, i.e. between observations. It is unrelated to what are the probabilistic/statistical relations between the random variables in each observation (in the general case we treat here where each observation is multidimensional).

Note also that in cases where we have continuous random variables with no densities, the above can be expressed in terms of the distribution functions.

This is what "independent observations" means. It is a precisely defined property expressed in mathematical terms. Let's see some of what it implies.

SOME CONSEQUENCES OF HAVING INDEPENDENT OBSERVATIONS

A. If two observations are part of a group of jointly independent observations, then they are also "pair-wise independent" (statistically),

$$f(\mathbb x_i,\mathbb x_m) = f_i(\mathbb x_i)f_m(\mathbb x_m)\;\;\; \forall i\neq m, \;\;\; i,m =1,...,n$$

This in turn implies that conditional PMF's/PDFs equal the "marginal" ones

$$f(\mathbb x_i \mid \mathbb x_m) = f_i(\mathbb x_i)\;\;\; \forall i\neq m, \;\;\; i,m =1,...,n$$

This generalizes to many arguments, conditioned or conditioning, say

$$f(\mathbb x_i , \mathbb x_{\ell}\mid \mathbb x_m) = f(\mathbb x_i , \mathbb x_{\ell}),\;\;\;\; f(\mathbb x_i \mid \mathbb x_m, \mathbb x_{\ell}) = f_i(\mathbb x_i)$$

etc, as long as the indexes to the left are different to the indexes on the right of the vertical line.

This implies that if we actually observe one observation, the probabilities characterizing any other observation of the sample do not change. So as regards prediction, an independent sample is not our best friend. We would prefer to have dependence so that each observation could help us say something more about any other observation.

B. On the other hand, an independent sample has maximum informational content. Every observation, being independent, carries information that cannot be inferred, wholly or partly, by any other observation in the sample. So the sum total is maximum, compared to any comparable sample where there exists some statistical dependence between some of the observations. But of what use is this information, if it cannot help us improve our predictions?

Well, this is indirect information about the probabilities that characterize the random variables in the sample. The more these observations have common characteristics (common probability distribution in our case), the more we are in a better position to uncover them, if our sample is independent.

In other words if the sample is independent and "identically distributed", meaning

$$f_i(\mathbb x_i) = f_m(\mathbb x_m) = f(\mathbb x),\;\;\; i\neq m$$

it is the best possible sample in order to obtain information about not only the common joint probability distribution $f(\mathbb x)$, but also for the marginal distributions of the random variables that comprise each observation, say $f_j(x_{ji})$.

So even though $f(\mathbb x_i \mid \mathbb x_m) = f_i(\mathbb x_i)$, so zero additional predictive power as regards the actual realization of $\mathbb x_i$, with an independent and identically distributed sample, we are in the best position to uncover the functions $f_i$ (or some of its properties), i.e. the marginal distributions.

Therefore, as regards estimation (which is sometimes used as a catch-all term, but here it should be kept distinct from the concept of prediction), an independent sample is our "best friend", if it is combined with the "identically distributed" property.

C. It also follows that an independent sample of observations where each is characterized by a totally different probability distribution, with no common characteristics whatsoever, is as worthless a collection of information as one can get (of course every piece of information on its own is worthy, the issue here is that taken together these cannot be combined to offer anything useful). Imagine a sample containing three observations: one containing (quantitative characteristics of) fruits from South America, another containing mountains of Europe, and a third containing clothes from Asia. Pretty interesting information pieces all three of them -but together as a sample cannot do anything statistically useful for us.

Put in another way, a necessary and sufficient condition for an independent sample to be useful, is that the observations have some statistical characteristics in common. This is why, in Statistics, the word "sample" is not synonymous to "collection of information" in general, but to "collection of information on entities that have some common characteristics".

APPLICATION TO THE OP'S DATA EXAMPLE

Responding to a request from user @gung, let's examine the OP's example in light of the above. We reasonably assume that we are in a school with more than two teachers and more than six pupils. So a) we are sampling both pupilss and teachers, and b) we include in our data set the grade that corresponds to each teacher-pupil combination.

Namely, the grades are not "sampled", they are a consequence of the sampling we did on teachers and pupils. Therefore it is reasonable to treat the random variable $G$ (=grade) as the "dependent variable", while pupils ($P$) and teachers $T$ are "explanatory variables" (not all possible explanatory variables, just some). Our sample consists of six observations which we write explicitly, $S = (\mathbb s_1, ..., \mathbb s_6)$ as

\begin{align} \mathbb s_1 =(T_1, P_1, G_1) \\ \mathbb s_2 =(T_1, P_2, G_2) \\ \mathbb s_3 =(T_1, P_3, G_3) \\ \mathbb s_3 =(T_2, P_4, G_4) \\ \mathbb s_4 =(T_2, P_5, G_5) \\ \mathbb s_5 =(T_2, P_6, G_6) \\ \end{align}

Under the stated assumption "pupils do not influence each other", we can consider the $P_i$ variables as independently distributed. Under a non-stated assumption that "all other factors" that may influence the Grade are independent of each other, we can also consider the $G_i$ variables to be independent of each other.
Finally under a non-stated assumption that teachers do not influence each other, we can consider the variables $T_1, T_2$ as statistically independent between them.

But irrespective of what causal/structural assumption we will make regarding the relation between teachers and pupils, the fact remains that observations $\mathbb s_1, \mathbb s_2, \mathbb s_3$ contain the same random variable ($T_1$), while observations $\mathbb s_4, \mathbb s_5, \mathbb s_6$ also contains the same random variable ($T_2$).

Note carefully the distinction between "the same random variable" and "two distinct random variables that have identical distributions".

So even if we assume that "teachers do NOT influence pupils", then still, our sample as defined above is not an independent sample, because $\mathbb s_1, \mathbb s_2, \mathbb s_3$ are statistically dependent through $T_1$, while $\mathbb s_4, \mathbb s_5, \mathbb s_6$ are statistically dependent through $T_2$.

Assume now that we exclude the random variable "teacher" from our sample. Is the (Pupil, Grade) sample of six observations, an independent sample? Here, the assumptions we will make regarding what is the structural relationship between teachers, pupils, and grades does matter.

First, do teachers directly affect the random variable "Grade", through perhaps, different "grading attitudes/styles"? For example $T_1$ may be a "tough grader" while $T_2$ may be not. In such a case "not seeing" the variable "Teacher" does not make the sample independent, because it is now the $G_1, G_2, G_3$ that are dependent, due to a common source of influence, $T_1$ (and analogously for the other three).

But say that teachers are identical in that respect. Then under the stated assumption "teachers influence students" we have again that the first three observations are dependent with each other, because teachers influence pupils who influence grades, and we arrive at the same result, albeit indirectly in this case (and likewise for the other three). So again, the sample is not independent.

THE CASE OF GENDER

Now, let's make the (Pupil, Grade) six-observation sample "conditionally independent with respect to teacher" (see other answers) by assuming that all six pupils have in reality the same teacher. But in addition let's include in the sample the random variable "$Ge$=Gender" that traditionally takes two values ($M,F$), while recently has started to take more. Our once again three-dimensional six-observation sample is now

\begin{align} \mathbb s_1 =(Ge_1, P_1, G_1) \\ \mathbb s_2 =(Ge_2, P_2, G_2) \\ \mathbb s_3 =(Ge_3, P_3, G_3) \\ \mathbb s_3 =(Ge_4, P_4, G_4) \\ \mathbb s_4 =(Ge_5, P_5, G_5) \\ \mathbb s_5 =(Ge_6, P_6, G_6) \\ \end{align}

Note carefully that what we included in the description of the sample as regards Gender, is not the actual value that it takes for each pupil, but the random variable "Gender". Look back at the beginning of this very long answer: the Sample is not defined as a collection of numbers (or fixed numerical or not values in general), but as a collection of random variables (i.e. of functions).

Now, does the gender of one pupil influences (structurally or statistically) the gender of the another pupil? We could reasonably argue that it doesn't. So from that respect, the $Ge_i$ variables are independent. Does the gender of pupil $1$, $Ge_1$, affects in some other way directly some other pupil ($P_2, P_3,...$)? Hmm, there are battling educational theories if I recall on the matter. So if we assume that it does not, then off it goes another possible source of dependence between observations. Finally, does the gender of a pupil influence directly the grades of another pupil? if we argue that it doesn't, we obtain an independent sample (conditional on all pupils having the same teacher).