What does orthogonal mean in the context of statistics? In other contexts, orthogonal means "at right angles" or "perpendicular".
What does orthogonal mean in a statistical context?
Thanks for any clarifications.
 A: I can't make a comment because I don't have enough points, so I'm forced to speak my mind as an answer, please forgive me.  From the little I know, I disagree with the selected answer by @crazyjoe because orthogonality is defined as 
$$E[XY^{\star}] = 0$$
So:
If $Y=X^2$ with symmetric pdf they they are dependent yet orthogonal.  
If $Y=X^2$ but pdf zero for negative values, then they dependent but not orthogonal.
Therefore, orthogonality does not imply independence.
A: It's most likely they mean 'unrelated' if they say 'orthogonal'; if two factors are orthogonal (e.g. in factor analysis), they are unrelated, their correlation is zero.
A: If X and Y are independent then they are Orthogonal. But the converse is not true as pointed out by the clever example of user497804. For the exact definitions refer to
Orthogonal : 
Complex-valued random variables $C_1$ and $C_2$ are called orthogonal if they satisfy ${\rm cov}(C_1,C_2)=0$
(Pg 376, Probability and Random Processes by Geoffrey Grimmett and David Stirzaker)
Independent:
The random variables $X$ and $Y$ are independent if and only if
$F(x,y) = F_X(x)F_Y(y)$ for all $x,y \in \mathbb{R}$
which, for continuous random variables, is equivalent to requiring that
$f(x,y) = f_X(x)f_Y(y)$
(Page 99, Probability and Random Processes by Geoffrey Grimmett and David Stirzaker)
A: @Mien already provided an answer, and, as pointed out by @whuber, orthogonal means uncorrelated. However, I really wish people would provide some references. You might consider the following links helpful since they explain the concept of correlation from a geometric perspective.

*

*The Geometry of Vectors (see p. 7)

*Linearly Independent, Orthogonal, and Uncorrelated Variables

*Graphical representation of two-dimensional correlation in vector space (may not be free to you)

A: According to https://web.archive.org/web/20160705135417/http://terpconnect.umd.edu/~bmomen/BIOM621/LineardepCorrOrthogonal.pdf, linear independency is a necessary condition for orthogonality or uncorrelatedness. But there are finer distinctions, in particular, orthogonality is not uncorrelatedness.
A: I asked a similar question What is the relationship between orthogonality and the expectation of the product of RVs, and I reproduce the answer here. Although orthogonality is a concept from Linear Algebra, and it means that the dot-product of two vectors is zero, the term is sometimes loosely used in statistics and means non-correlation. If two random vectors are orthogonal, then their centralized counterpart are uncorrelated, because orthogonality (dot-product zero) implies non-correlation of the centralized random vectors (sometimes people say that orthogonality implies that the cross-moment is zero). Whenever we have two Random Vectors $(X,Y)$, we can always centralize them around their means to make their expectation to be zero. Assume ortogonality ($X\cdot Y=0$), then the correlation of the centralized random variables are
$$Cov(X-E[X],Y-E[Y]) = E[X\cdot Y]= E[0]=0\implies \\Corr(X-E[X],Y-E[Y])=0$$
A: A NIST website (ref below) defines orthogonal as follows, "An experimental design is orthogonal if the effects of any factor balance out (sum to zero) across the effects of the other factors." 
In statistical deisgn, I understand orthogonal to mean "not cofounded" or "not aliased". This is important when designing and analyzing your experiment if you want to make sure you can clearly identify different factors/treatments. If your designed experiment is not orthogonal, then it means you will not be able to completely separate the effects of different treatments. Thus you will need to conduct a follow up experiment to deconfound the effect. This would be called augmented deisgn or comparitive design.
Independence seems to be a poor word choice since its used in so many other aspects of design and analysis.
NIST Ref
http://www.itl.nist.gov/div898/handbook/pri/section7/pri7.htm
A: In econometrics, the orthogonality assumption means the expected value of the sum of all errors is 0. All variables of a regressor is orthogonal to their current error terms.
Mathematically, the orthogonality assumption is $E(x_{i}·ε_{i}) = 0$.
In simpler terms, it means a regressor is "perpendicular" to the error term.
A: Assume a random process x(t), hence y1=cos(x(t)) and y2= sin(x(t)), both are random processes. It is clear that y1 is orthogonal on y2, i.e., E[y1.y2] = 0. However, indeed they are dependent on each other. Actually, both are based on the same random process. Therefore, it is not necessary for orthogonal processes to be independent. Independence in random processes means that if you have any foreknowledge about one process, you will not be able to have any conclusion about the other! However, this is not the case with orthogonal processes. Nevertheless, assume two independent random processes z1, z2  where at least one of them has zero mean, then E[z1.z2]=E[z1].E[z2]=0. Mathematically, this is the same as the orthogonality condition, but geometrically, it is not necessary!
A: Two or more IV's unrelated (independent) to one another but both having an influence on the DV. Each IV  separately contributes a distinct value to the outcome , while both or all IV's also contribute in an additive fashion in the prediction of income (orthogonal=non-intersecting IV's influence on a DV). IV's are non-correlational amongst one another and usually positioned in a right angle *see Venn Diagram.
Example: Relationship among motivation and years of education on income.
IV= Years of Education
IV= Motivation
DV= Income
https://web.archive.org/web/20160216160117/https://onlinecourses.science.psu.edu/stat505/node/167
A: The related random variables mean the variables say X and Y can have any relationship; may be linear or non-linear. The independence and orthogonal properties are the same if the two variables are linearly related.
A: It means they [the random variables X,Y] are 'independent' to each other.  Independent random variables are often considered to be at 'right angles' to each other, where by 'right angles' is meant that the inner product of the two is 0 (an equivalent condition from linear algebra).  
For example on the X-Y plane the X and Y axis are said to be orthogonal because if a given point's x value changes, say going from (2,3) to (5,3), its y value remains the same (3), and vice versa.  Hence the two variables are 'independent'. 
See also Wikipedia's entries for Independence and Orthogonality
