Rieppel's reptile matrix and the turtle problem

Which group turtles are most closely related to is one of the greatest problems in vertebrate paleontology.  Traditionally parareptiles, deBraga and Rieppel (1997) and variations on that matrix find them to be close to Sauropterygia in Lepidosauromorpha, while molecular analyses find them to be archosauromorphs.  Recently, Lyson et al. (2010) added Proganochelys and Eunotosaurus to the Rieppel matrix, along with six characters they share,  and found turtles now clade with it inside Parareptilia instead.  What Lyson et al. don't mention is that diapsid turtles are only seven steps longer.  Not much of a difference, but it's something, right?  I'm not so sure.  Looking at the characters, there are huge problems.  I give Peters a lot of (deserved) flack for the poorly designed characters in his amniote analysis, but this honestly isn't any better.  Let's examine some problems...

Composite Characters
Many characters describe more than one variable in morphology, which is not useful.  These can have states describing different variables (e.g. teeth serrated vs. teeth large) or multiple variables described in a single state (e.g. teeth large and serrated vs. teeth small and unserrated).  A common variant is to have "x feature absent" as a state of a character which otherwise describes different morphologies (e.g. teeth serrated vs. teeth unserrated vs. teeth absent).  The only way an absent state should be in a character that's not simply "absent vs. present" is if the variable is number (e.g. two centrales vs. one centrale vs. no centrale) or size (e.g. quadratojugal large vs. quadratojugal small vs. quadratojugal absent).  Characters with this issue are-

1. Premaxilla exposure: exposure anterolateral to external nares small restricted to low posterolateral process forming less than one-half the height of the premaxilla (0); posterolateral process tall reaching dorsal process (1).
6. External nares exposure: dorsal process of premaxilla broad restricting nares to a lateral exposure (0); dorsal process narrow resulting in dorsal exposure of nares (1).
8. Choana palatal exposure: parallel medial border of maxilla (0); deflected posteromedially (1); hidden in palatal view (2).
17. Lacrimal morphology: present and contributing to exteral nares (0); present at least as long as tall, but excluded from external nares (1); if present snall[sic], restricted to orbital margin, or absent entirely (2).
30. Postorbital/supratemporal relationship: in contact (0); not in contact (1); supratemporal absent (1).
38. Quadrate excavation: absent along posterior edge (0); posterior edge deeply excavated forming a concave region (1); quadrate greatly reduced (2).
42. Quadratojugal morphology: present and horizontal dimension exceeds vertical dimension by a factor of at least three (0); present but vertical dimension exceeds horizontal by a factor of at least two (1); present, but greatly reduced and restricted to condylar region (2); absent (3).
49. Pineal foramen position: located in the middle of the body from the parietal (0); displaced posteriorly (1); displaced anteriorly (2); absent (3).
51. Lower temporal fenestra: absent (0); present quadratojugal included (1); present quadratojugal excluded (2); open ventrally (3).
52. Postparietal: present and paired (0); present but fused (1); absent (2).
60. Orientation of paroccipital process: extends laterally forming 90° with parasaggital plane (0); paroccipital process deflected posterolaterally at an angle of about 20° from the transverse width of the skull (1); paroccipital process deflected dorsolaterally at an angle of nearly 45° (2).
65. Ventral braincase tubera: absent (0); present and restricted to basioccipital (1); present, very large, and restricted to basisphenoid (2).
73. Interpterygoid vacuity: anterior end tapers sharply (0); anterior border cresentric (1); absent (2).
74. Suborbital fenestra: absent (0); present but with contribution from either maxilla or jugal along lateral border (1); present, but with both maxilla and jugal excluded from lateral border (2).
75. Cultriform process: long, exceeding length of parasphenoid body and reaching forward to the level of the posterior limit of the internal nares (0); short, not reaching the level of the internal nares (1).
81. Ectopterygoid: present and edentulous (0); present and dentigerous (1); absent replaced by medial process of jugal (2); absent replaced by lateral process of pterygoid (3).
83. Coronoid process: absent (0); present formed by coronoid (1); present formed by dentary (2).
94. Tooth implantation: set in deep sockets ( ); loosely attached to medial surface of jaw (1); ankylosed to jaw (2).
100. Vertebral central articulations: amphicoelous (0); platycoelous (1); other (2).
106. Trunk neural arches: swollen with heavy zygapophyseal butress (0); narrow, strongly excavated neural arch with no heavy butress (1); swollen, but with narrow tall zygapophyseal butress (2).
118. Acromion process: absent (0); present, blade-like, parallelogram in lateral aspect, and arising from the lateral edge of the scapula (1); present, triangular in lateral aspect, and arising from ventromedial border of scapula (2).
126. Supinator process: large angled away from humeral shaft (0); large confluent with shaft (1); small or absent (2).
136. Pubic tubercle: if present small and directed anteroventrally (0); large and strongly turned ventrally (1).
139. Femoral shaft: short and stout (0); sigmoidally curved and slender (1).
141. Femoral trochanter major: absent (0); present and deflected distally from the proximal head of the femur (0); pyramidal in shape and nearly in line with the head of the femur (2); similar in shape to state (1) but positioned at mid-shaft length (3).
149. Astragalus/calcaneum relationship in adult: never fused (0); fused (1); hinge present (2).
159. Fifth pedal digit: longer than first digit (0); shorter and more lightly built than first (1).

Lyson et al.'s new character 176 is also a composite-

176. Dermal skull tuberosities: absent (0); tuberosities present (1); tuberosities and pits present (2); honeycomb texture present (3).

Unordered Characters
All characters were ran unordered, but some kinds of characters need to be ordered, when one state is intermediate.  Sometimes the intermediate state isn't placed between the others, which means the states have to be switched and recoded in the matrix for the ordering to work.  Other times, some of the states should be ordered, but others don't belong in the same character since they describe different variables (as noted above).  These can only be fixed by dividing the character.  Characters with this issue are-

19. Skull proportions: preorbital skull length equal to postorbital length (0); preorbital length exceeds postorbital skull length (1); postorbital length exceeds preorbital skull length (2).
35. Squamosal lateral exposure: ventral process long, descends to level limit of orbital margin (0); ventral process short, terminates prior to reaching ventral orbital margin (1); ventral process absent or restricted to region above dorsal limit of orbit (2).
42. Quadratojugal morphology: present and horizontal dimension exceeds vertical dimension by a factor of at least three (0); present but vertical dimension exceeds horizontal by a factor of at least two (1); present, but greatly reduced and restricted to condylar region (2); absent (3).
47. Parietal skull table: broad with the mid-line, transverse, width not less than half of the length measured along the element’s midline (0); constricted with the length exceeding the width by at least three times (1); forming saggital crest (2).
49. Pineal foramen position: located in the middle of the body from the parietal (0); displaced posteriorly (1); displaced anteriorly (2); absent (3).
53. Supratemporal: present and large with its transverse dimension nearly equal to its parasaggital dimension (0); present but reduced so that its transverse dimension is less than half of its parasaggital dimension (1); absent (2).
55. Tabular: present but restricted to dorsal region of occiput (0); present but ventrally elongate descending to lvel[sic] of occipital condyle (1); absent (2).
59. Posttemporal fenestra: absent (0); present but diameter less than half of the diameter of the foramen magnum (1); large posttemporal fenestra with a diameter at least eqqual[sic] to that of the foramen magnum (2).
76. Palatal process of pterygoid: extends anterior to the anterior limit of the palatine (0); forms oblique suture with palatine but process ends before reaching anterior limit of palatine (1); forms transverse suture with palatine (2).
78. Dentition on transverse flange: present as a shagreen of teeth (0); present but with one large distinct row of teeth along the posterior edge of the transverse flange (1); edentulous (2).
82. Mandibular joint: even with occiput (0); behind occiput (1); anterior to occiput (2).
88. Splenial: enters mandibular symphysis (0); present but excluded from mandibular symphysis (1); entirely absent (2).
89. Angular lateral exposure: exposed along 1/3 the lateral face of the mandible (0); exposed only as a small sliver along the lateral face (1); absent from lateral aspect (2).
100. Vertebral central articulations: amphicoelous (0); platycoelous (1); other (2).
115. Interclavicle: anterior end rhomboidal (0); T-shaped but with broad transverse bar with its anteroposterior dimension at least 1/4 the transverse width of the bar (1); T-shaped but transverse bar slender with its anteroposterior dimension much less than 1/4 the transverse width (2).
117. Scapula: short and broad with its height not exceeding its width (measured at the level of the glenoid) by more than three times (0); tall and blade-like with its height exceeding the width by at least a factor of four (1); tall and slender nearly cylindrical in cross-section (2).
126. Supinator process: large angled away from humeral shaft (0); large confluent with shaft (1); small or absent (2).
129. Radius/ulna ratio: radius shorter than ulna (0); radius longer than ulna (1); radius and ulna subequal (2).
142. Intertrochanteric fossa: well defined (0); reduced (1); absent (2).
150. Astragalus/distal tarsal IV articularion: articulation poorly defined (0); articulation well defined (1); articulation absent (2).
158. Number of pedal centralia: both lateral and medial centralia present (0); medial pedal centralia lost (1); both centralia lost (2).
161. Pedal phalangeal formula: 2, 3, 4, 5(4), 4 (0); 2, 3, 4, 4, 3 (1); 2, 3, 3, 4, 3 or less (2).
165. Body osteoderms: absent (0); present but few restricted to mid-line (1); present but spread all over back (2).
166. Osteodermal ridges: absent (0); fine regular spaced ridges (2); heavy irregularly spaced ridges (3).

Lyson et al.'s new character 176 also has states that should be ordered once it is divided into 2 separate characters-

176. Dermal skull tuberosities: absent (0); tuberosities present (1); tuberosities and pits present (2); honeycomb texture present (3).

Characters Which Do Not Cover All Possibilities
The character states must combine to cover the possible range of morphologies, but sometimes gaps are left between them, which makes coding taxa within that gap problematic.  Examples of this are-

20. Prefrontal/palatine antorbital contact: narrow forming less than 1/3 the transverse distance between the orbits (0); contact broad forming at least 1/2 the distance between the orbits (1).
26. Frontal proportions: length exceeds width by at least four times (0); length no greater than twice the width (1).
27. Frontal morphology: parallelogram shaped (0); hour-glass shaped (1).
42. Quadratojugal morphology: present and horizontal dimension exceeds vertical dimension by a factor of at least three (0); present but vertical dimension exceeds horizontal by a factor of at least two (1); present, but greatly reduced and restricted to condylar region (2); absent (3).
45. Stapes morphology: robust with its greatest depth exceeding one-third of its total length (0); slender with the length at least four times the depth (1).
47. Parietal skull table: broad with the mid-line, transverse, width not less than half of the length measured along the element’s midline (0); constricted with the length exceeding the width by at least three times (1); forming saggital crest (2).
59. Posttemporal fenestra: absent (0); present but diameter less than half of the diameter of the foramen magnum (1); large posttemporal fenestra with a diameter at least eqqual[sic] to that of the foramen magnum (2).
61. Paroccipital process morphology: slender with anteroposterior dimension not exceeding dorsoventral dimension (0); heavy with anteroposterior dimension at least 1/3 greater than dorsoventral dimension (1).
64. Basi/parasphenoid ratio: narrowest transverse width no more than 60% of the maximum length measured from basipterygoid process to posteriomost limit (0); narrowest part (waist) exceeds 80% of the length (1).
114. Clavicle: interclavicular process of clavicle broad and blade-like with the maximum anteroposterior length at least 1/3 of its transverse dimension (0); slender with its anteroposterior length less than 1/5 of the transverse dimension (1).
124. Humeral shaft/distal end ratio: shaft length less than 1/3 the maximum width of the distal end of the humerus (0); shaft long at least four times the width of the distal end (1).
154. Metatarsal V: long and slender with length exceeding the width of the base by at least three times (0); short and broad with base width equivalent to at least twice the length of the element measured along its midline (1).

Poorly Defined Characters
These are too numerous to mention, but many characters lack a quantification.  Some are especially bad, like "Limbs: short and stout (0); long and slender (1)." 

The latter two kinds of characters don't really affect the matrix, they just make coding new taxa problematic since you could have a different concept of "stout" or "large" or how curved or angled something has to be to count.  Who knows if Li et al.'s coding of Odontochelys or Lyson et al.'s coding of Proganochelys and Eunotosaurus used the same concepts as deBraga and Rieppel's original codings?  Even considering only the first two kinds of character errors though, 47 of the 168 characters are affected.  That's 28%. 

Ordering the (non-composite) characters is easy enough.  After deleting Testudines (since Proganochelys was used as a major source, and is now its own OTU), it takes 7 more steps to place turtles in Diapsida in Lyson et al.'s original matrix.  After proper ordering, the topology is the same except kuehneosaurs are now sister to a clade of lepidosaurs and sauropterygians instead of in a trictomy with both groups.  It now takes 8 more steps for diapsid turtles.  After proper ordering and deleting the composite characters, the consensus tree differs in that Macroleter is closer to derived parareptiles than lanthanosuchids and Acleistorhinus, and Sauria breaks down except for Trilophosaurus+Rhynchosauria and sauropterygians, with prolacertiforms oddly basal.  It now only takes three steps to place turtles in Diapsida though.  In all of these analyses, they are sister to sauropterygians when in Diapsida.

Factor in the small number of characters and taxa, and my conclusion is that amniote matrices aren't good enough to tell us much about turtle relationships yet.