r : provides a measure of the sensitivity to electron supply or withdrawal. It may be (+) or (-).

If r = (+); the reaction is accelerated by electron withdrawing substituents.

If r = (-); the reaction is accelerated by electron donating substituents.

The magnitude of r:  r is highly dependent on the substituent distance from the reaction center. It may be considered as an effect of insulation. The slope intensity and direction of the line also can reveal changes or indications of mechanism. In biological systems the value may show where binding [to the molecule] occurs.   Consider first, r -values that are positive but differ in their magnitude. The effect of insulation has been shown to correlate quite well with the r value.  Specifically,

The ionization of organic acids (in the graph below; D = r and F = s)
 

Consider the following application of QSAR to serine protease inhibitors of the O-aryl-diethylphosphates class. O-Aryl-diethylphosphates react with a serine-OH to form a phosphorylated [inhibited] protein that is rendered unable to participate in the hydrolysis of the natural substrate.  Synchronous with the phosphorylation is the ejection (displacement) of the substituted O-aryl moiety (Scheme 1).  How do substituents on the O-aryl group effect the reactivity or inhibitory potency of this class of compounds?   What is the mechanism?   Is the phosphate protonated (general acid catalysis; + charge in transition state), followed by neutral nucleophilic attack, or is the serine hydroxyl ionized to nucleophilic Ser-O^- that reacts with the phosphate ester (general base catalysis; - charge formed in transition state). A study of the substituent effect would reveal this difference. As a guide to the inhibitory action, simple hydrolysis with OH^- also was conducted.

Change in Sign of r!

Consider amide hydrolysis by base (amide bonds make up the backbone of proteins = peptide bond) (Eqn. 1).

Next, consider amide hydrolysis by acids (Eqn. 2).

Why is there a change in sign for the two hydrolysis procedures? The mechanisms differ - base hydrolysis involves attack at the carbonyl by an OH- thereby producing a negative charge in the transition state or intermediate step. Electron withdrawing groups favor this step and therefore the value will be positive. The acid hydrolysis mechanism involves preliminary protonation of the carbonyl group by an H+ to produce a delocalized cation in the intermediate step. Electron donating groups favor this step thereby producing a negative value. All of this occurs because the overall extent and efficacy of the reaction is determined in the rate-determining step - the activation energy needed to overcome the barrier of a reaction. For example, when a positive charge is produced in a transition state or intermediate, electron donors stabilize this charge by pushing electrons toward the electron deficient site. The net result is that electron donors stabilize cations, reduce the energy and make the hill (barrier) lower to overcome. The same goes for electron withdrawing groups and anions (negative charges).
 

Back to Group Effects:  Remember some groups act by resonance. These include:

        Electron withdrawers by resonance: CO₂H, CN, C(O)R, S(O)R, SO₂R

        Electron donors by resonance: OR, SR, X (X=Cl,Br,I), OH, NH₂, NHR
 

e.g.,

And their sigma values may reflect both inductive and resonance interactions.

The difference between these values gives an indication of the resonance contribution. For s_p+, the r is usually negative!

Same basic idea as sigma values (described by Taft). Steric contribution = Es

Es is defined as the difference between the logarithm of the relative rate of acid-catalyzed hydrolysis of a [substituted] carboxy methyl ester and the log of the rate of hydrolysis of methyl acetate (standard).

The rate of hydrolysis for both acid and base-promoted reactions was studied and the realtive rates compared to achieve the Es values.
A summary of some important Es values will enable you to grasp the relative "size" of alkyl and functional groups in a way better than the DG values.  In this series, methyl is the standard substituent (rather than H).